2004
2004
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1.I.1B
2004 commentThe linear map represents reflection in the plane through the origin with normal , where , and referred to the standard basis. The map is given by , where is a matrix.
Show that
Let and be unit vectors such that is an orthonormal set. Show that
and find the matrix which gives the mapping relative to the basis .
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1.I.2C
2004 commentShow that
for any real numbers . Using this inequality, show that if and are vectors of unit length in then .
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1.II.5B
2004 commentThe vector satisfies the equation
where is a matrix and is a column vector. State the conditions under which this equation has (a) a unique solution, (b) an infinity of solutions, (c) no solution for .
Find all possible solutions for the unknowns and which satisfy the following equations:
in the cases (a) , and (b) .
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1.II.6A
2004 commentExpress the product in suffix notation and thence prove that the result is zero.
Silver Beard the space pirate believed people relied so much on space-age navigation techniques that he could safely write down the location of his treasure using the ancient art of vector algebra. Spikey the space jockey thought he could follow the instructions, by moving by the sequence of vectors one stage at a time. The vectors (expressed in 1000 parsec units) were defined as follows:
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Start at the centre of the galaxy, which has coordinates .
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Vector a has length , is normal to the plane and is directed into the positive quadrant.
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Vector is given by , where .
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Vector has length , is normal to and , and moves you closer to the axis.
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Vector .
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Vector has length . Spikey was initially a little confused with this one, but then realised the orientation of the vector did not matter.
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Vector has unknown length but is parallel to and takes you to the treasure located somewhere on the plane .
Determine the location of the way-points Spikey will use and thence the location of the treasure.
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1.II.7A
2004 commentSimplify the fraction
where is the complex conjugate of . Determine the geometric form that satisfies
Find solutions to
and
where is a complex variable. Sketch these solutions in the complex plane and describe the region they enclose. Derive complex equations for the circumscribed and inscribed circles for the region. [The circumscribed circle is the circle that passes through the vertices of the region and the inscribed circle is the largest circle that fits within the region.]
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1.II.8C
2004 comment(i) The vectors in satisfy . Are necessarily linearly independent? Justify your answer by a proof or a counterexample.
(ii) The vectors in have the property that every subset comprising of the vectors is linearly independent. Are necessarily linearly independent? Justify your answer by a proof or a counterexample.
(iii) For each value of in the range , give a construction of a linearly independent set of vectors in satisfying
where is the Kronecker delta.
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3.I.1D
2004 commentState Lagrange's Theorem.
Show that there are precisely two non-isomorphic groups of order 10 . [You may assume that a group whose elements are all of order 1 or 2 has order .]
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3.II.7D
2004 commentLet be a real symmetric matrix. Show that all the eigenvalues of are real, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal to each other.
Find the eigenvalues and eigenvectors of
Give an example of a non-zero complex symmetric matrix whose only eigenvalues are zero. Is it diagonalisable?
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3.I.2D
2004 commentDefine the Möbius group, and describe how it acts on .
Show that the subgroup of the Möbius group consisting of transformations which fix 0 and is isomorphic to .
Now show that the subgroup of the Möbius group consisting of transformations which fix 0 and 1 is also isomorphic to .
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3.II.5D
2004 commentLet be the dihedral group of order 12 .
i) List all the subgroups of of order 2 . Which of them are normal?
ii) Now list all the remaining proper subgroups of . [There are of them.]
iii) For each proper normal subgroup of , describe the quotient group .
iv) Show that is not isomorphic to the alternating group .
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3.II.6D
2004 commentState the conditions on a matrix that ensure it represents a rotation of with respect to the standard basis.
Check that the matrix
represents a rotation. Find its axis of rotation .
Let be the plane perpendicular to the axis . The matrix induces a rotation of by an angle . Find .
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3.II.8D
2004 commentCompute the characteristic polynomial of
Find the eigenvalues and eigenvectors of for all values of .
For which values of is diagonalisable?
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1.I.3D
2004 commentDefine the supremum or least upper bound of a non-empty set of real numbers.
State the Least Upper Bound Axiom for the real numbers.
Starting from the Least Upper Bound Axiom, show that if is a bounded monotonic sequence of real numbers, then it converges.
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1.I.4E
2004 commentLet for . Show by induction or otherwise that for every integer ,
Evaluate the series
[You may use Taylor's Theorem in the form
without proof.]
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1.II.9D
2004 commenti) State Rolle's theorem.
Let be continuous functions which are differentiable on .
ii) Prove that for some ,
iii) Suppose that , and that exists and is equal to .
Prove that exists and is also equal to .
[You may assume there exists a such that, for all and
iv) Evaluate .
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1.II.10E
2004 commentDefine, for an integer ,
Show that for every , and deduce that
Show that , and that
Hence prove that
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1.II.11F
2004 commentLet be defined on , and assume that there exists at least one point at which is continuous. Suppose also that, for every satisfies the equation
Show that is continuous on .
Show that there exists a constant such that for all .
Suppose that is a continuous function defined on and that, for every , satisfies the equation
Show that if is not identically zero, then is everywhere positive. Find the general form of .
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1.II.12F
2004 comment(i) Show that if and
for all , and if converges, then converges.
(ii) Let
By considering , or otherwise, show that as .
[Hint: for .]
(iii) Determine the convergence or otherwise of
for (a) , (b) .
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2.I.1B
2004 commentBy writing where is a constant, solve the differential equation
and find the possible values of .
Describe the isoclines of this differential equation and sketch the flow vectors. Use these to sketch at least two characteristically different solution curves.
Now, by making the substitution or otherwise, find the solution of the differential equation which satisfies .
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2.I.2B
2004 commentFind two linearly independent solutions of the differential equation
Find also the solution of
that satisfies
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2.II.5B
2004 commentConstruct a series solution valid in the neighbourhood of , for the differential equation
satisfying
Find also a second solution which satisfies
Obtain an expression for the Wronskian of these two solutions and show that
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2.II.6B
2004 commentTwo solutions of the recurrence relation
are given as and , and their Wronskian is defined to be
Show that
Suppose that , where is a real constant lying in the range , and that . Show that two solutions are and , where . Evaluate the Wronskian of these two solutions and verify .
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2.II.7B
2004 commentShow how a second-order differential equation may be transformed into a pair of coupled first-order equations. Explain what is meant by a critical point on the phase diagram for a pair of first-order equations. Hence find the critical points of the following equations. Describe their stability type, sketching their behaviour near the critical points on a phase diagram.
Sketch the phase portraits of these equations marking clearly the direction of flow.
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2.II.8B
2004 commentConstruct the general solution of the system of equations
in the form
and find the eigenvectors and eigenvalues .
Explain what is meant by resonance in a forced system of linear differential equations.
Consider the forced system
Find conditions on and such that there is no resonant response to the forcing.
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4.I.3A
2004 commentA lecturer driving his car of mass along the flat at speed accidentally collides with a stationary vehicle of mass . As both vehicles are old and very solidly built, neither suffers damage in the collision: they simply bounce elastically off each other in a straight line. Determine how both vehicles are moving after the collision if neither driver applied their brakes. State any assumptions made and consider all possible values of the mass ratio . You may neglect friction and other such losses.
An undergraduate drives into a rigid rock wall at speed . The undergraduate's car of mass is modern and has a crumple zone of length at its front. As this zone crumples upon impact, it exerts a net force on the car, where is the amount the zone has crumpled. Determine the value of at the point the car stops moving forwards as a function of , where .
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4.I.4A
2004 commentA small spherical bubble of radius a containing carbon dioxide rises in water due to a buoyancy force , where is the density of water, is gravitational attraction and is the volume of the bubble. The drag on a bubble moving at speed is , where is the dynamic viscosity of water, and an accelerating bubble acts like a particle of mass , for some constant . Find the location at time of a bubble released from rest at and show the bubble approaches a steady rise speed
Under some circumstances the carbon dioxide gradually dissolves in the water, which leads to the bubble radius varying as , where is the bubble radius at and is a constant. Under the assumption that the bubble rises at speed given by , determine the height to which it rises before it disappears.
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4.II.9A
2004 commentA horizontal table oscillates with a displacement , where is the amplitude vector and the angular frequency in an inertial frame of reference with the axis vertically upwards, normal to the table. A block sitting on the table has mass and linear friction that results in a force , where is a constant and is the velocity difference between the block and the table. Derive the equations of motion for this block in the frame of reference of the table using axes on the table parallel to the axes in the inertial frame.
For the case where , show that at late time the block will approach the steady orbit
where
and is a constant.
Given that there are no attractive forces between block and table, show that the block will only remain in contact with the table if .
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4.II.10A
2004 commentA small probe of mass is in low orbit about a planet of mass . If there is no drag on the probe then its orbit is governed by
where is the location of the probe relative to the centre of the planet and is the gravitational constant. Show that the basic orbital trajectory is elliptical. Determine the orbital period for the probe if it is in a circular orbit at a distance from the centre of the planet.
Data returned by the probe shows that the planet has a very extensive but diffuse atmosphere. This atmosphere induces a drag on the probe that may be approximated by the linear law , where is the drag force and is a constant. Show that the angular momentum of the probe about the planet decays exponentially.
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4.II.11A
2004 commentA particle of mass and charge moves through a magnetic field . There is no electric field or external force so that the particle obeys
where is the location of the particle. Prove that the kinetic energy of the particle is preserved.
Consider an axisymmetric magnetic field described by in cylindrical polar coordinates . Determine the angular velocity of a circular orbit centred on .
For a general orbit when , show that the angular momentum about the -axis varies as , where is the angular momentum at radius . Determine and sketch the relationship between and . [Hint: Use conservation of energy.] What is the escape velocity for the particle?
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4.II.12A
2004 commentA circular cylinder of radius , length and mass is rolling along a surface. Show that its moment of inertia is given by .
At the cylinder is at the bottom of a slope making an angle to the horizontal, and is rolling with velocity and angular velocity . Assuming slippage does not occur, determine the position of the cylinder as a function of time. What is the maximum height that the cylinder reaches?
The frictional force between the cylinder and surface is given by , where is the friction coefficient. Show that the cylinder begins to slip rather than roll if . Determine as a function of time the location, speed and angular velocity of the cylinder on the slope if this condition is satisfied. Show that slipping continues as the cylinder ascends and descends the slope. Find also the maximum height the cylinder reaches, and its speed and angular velocity when it returns to the bottom of the slope.
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4.I.1E
2004 comment(a) Use Euclid's algorithm to find positive integers such that .
(b) Determine all integer solutions of the congruence
(c) Find the set of all integers satisfying the simultaneous congruences
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4.I.2E
2004 commentProve by induction the following statements:
i) For every integer ,
ii) For every integer is divisible by 6 .
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4.II.5E
2004 commentShow that the set of all subsets of is uncountable, and that the set of all finite subsets of is countable.
Let be the set of all bijections from to , and let be the set
Show that is uncountable, but that is countable.
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4.II.6E
2004 commentProve Fermat's Theorem: if is prime and then .
Let and be positive integers with . Show that if where is prime and , then
Now assume that is a product of distinct primes. Show that if and only if, for every prime divisor of ,
Deduce that if every prime divisor of satisfies , then for every with , the congruence
holds.
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4.II.7E
2004 commentPolynomials for are defined by
Show that for every , and that if then .
Prove that if is any polynomial of degree with rational coefficients, then there are unique rational numbers for which
Let . Show that
Show also that, if and are polynomials such that , then is a constant.
By induction on the degree of , or otherwise, show that if for every , then for all .
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4.II.8E
2004 commentLet be a finite set, subsets of and . Let be the characteristic function of , so that
Let be any function. By considering the expression
or otherwise, prove the Inclusion-Exclusion Principle in the form
Let be an integer. For an integer dividing let
By considering the sets for prime divisors of , show that
(where is Euler's function) and
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2.I.3F
2004 commentDefine the covariance, , of two random variables and .
Prove, or give a counterexample to, each of the following statements.
(a) For any random variables
(b) If and are identically distributed, not necessarily independent, random variables then
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2.I.4F
2004 commentThe random variable has probability density function
Determine , and the mean and variance of .
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2.II.9F
2004 commentLet be a positive-integer valued random variable. Define its probability generating function . Show that if and are independent positive-integer valued random variables, then .
A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, ) and . Show that there exists a non-standard pair of dice and such that when thrown
total shown by and is total shown by pair of ordinary dice is
for all .
[Hint:
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2.II.10F
2004 commentDefine the conditional probability of the event given the event .
A bag contains four coins, each of which when tossed is equally likely to land on either of its two faces. One of the coins shows a head on each of its two sides, while each of the other three coins shows a head on only one side. A coin is chosen at random, and tossed three times in succession. If heads turn up each time, what is the probability that if the coin is tossed once more it will turn up heads again? Describe the sample space you use and explain carefully your calculations.
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2.II.11F
2004 commentThe random variables and are independent, and each has an exponential distribution with parameter . Find the joint density function of
and show that and are independent. What is the density of ?
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2.II.12F
2004 commentLet be events such that for . Show that the number of events that occur satisfies
Planet Zog is a sphere with centre . A number of spaceships land at random on its surface, their positions being independent, each uniformly distributed over the surface. A spaceship at is in direct radio contact with another point on the surface if . Calculate the probability that every point on the surface of the planet is in direct radio contact with at least one of the spaceships.
[Hint: The intersection of the surface of a sphere with a plane through the centre of the sphere is called a great circle. You may find it helpful to use the fact that random great circles partition the surface of a sphere into disjoint regions with probability one.]
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3.I.3C
2004 commentIf and are differentiable vector fields, show that
(i) ,
(ii) .
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3.I.4C
2004 commentDefine the curvature, , of a curve in .
The curve is parametrised by
Obtain a parametrisation of the curve in terms of its arc length, , measured from the origin. Hence obtain its curvature, , as a function of .
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3.II.9C
2004 commentFor a function state if the following implications are true or false. (No justification is required.)
(i) is differentiable is continuous.
(ii) and exist is continuous.
(iii) directional derivatives exist for all unit vectors is differentiable.
(iv) is differentiable and are continuous.
(v) all second order partial derivatives of exist .
Now let be defined by
Show that is continuous at and find the partial derivatives and . Then show that is differentiable at and find its derivative. Investigate whether the second order partial derivatives and are the same. Are the second order partial derivatives of at continuous? Justify your answer.
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3.II.10C
2004 commentExplain what is meant by an exact differential. The three-dimensional vector field is defined by
Find the most general function that has as its differential.
Hence show that the line integral
along any path in between points and vanishes for any values of and .
The two-dimensional vector field is defined at all points in except by
is not defined at .) Show that
for any closed curve in that goes around anticlockwise precisely once without passing through .
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3.II.11C
2004 commentLet be the 3 -dimensional sphere of radius 1 centred at be the sphere of radius centred at and be the sphere of radius centred at . The eccentrically shaped planet Zog is composed of rock of uniform density occupying the region within and outside and . The regions inside and are empty. Give an expression for Zog's gravitational potential at a general coordinate that is outside . Is there a point in the interior of where a test particle would remain stably at rest? Justify your answer.
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3.II.12C
2004 commentState (without proof) the divergence theorem for a vector field with continuous first-order partial derivatives throughout a volume enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface .
By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field
defined within a sphere of radius centred at the origin.
Suppose that functions are continuous and that their first and second partial derivatives are all also continuous in a region bounded by a smooth surface .
Show that
Hence show that if is a continuous function on and a continuous function on and and are two continuous functions such that
then for all in .
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1.I.4G
2004 commentDefine what it means for a sequence of functions , where , to converge uniformly to a function .
For each of the following sequences of functions on , find the pointwise limit function. Which of these sequences converge uniformly? Justify your answers.
(i)
(ii)
(iii)
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1.II.15G
2004 commentState the axioms for a norm on a vector space. Show that the usual Euclidean norm on ,
satisfies these axioms.
Let be any bounded convex open subset of that contains 0 and such that if then . Show that there is a norm on , satisfying the axioms, for which is the set of points in of norm less than 1 .
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2.I.3G
2004 commentConsider a sequence of continuous functions . Suppose that the functions converge uniformly to some continuous function . Show that the integrals converge to .
Give an example to show that, even if the functions and are differentiable, the derivatives need not converge to .
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2.II.14G
2004 commentLet be a non-empty complete metric space. Give an example to show that the intersection of a descending sequence of non-empty closed subsets of , can be empty. Show that if we also assume that
then the intersection is not empty. Here the diameter is defined as the supremum of the distances between any two points of a set .
We say that a subset of is dense if it has nonempty intersection with every nonempty open subset of . Let be any sequence of dense open subsets of . Show that the intersection is not empty.
[Hint: Look for a descending sequence of subsets , with , such that the previous part of this problem applies.]
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3.I.4F
2004 commentLet and be metric spaces with metrics and . If and are any two points of , prove that the formula
defines a metric on . If , prove that the diagonal of is closed in .
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2004
commentState and prove the contraction mapping theorem.
Let be a positive real number, and take . Prove that the function
is a contraction from to . Find the unique fixed point of .
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4.I.3F
2004 commentLet be open sets in , respectively, and let be a map. What does it mean for to be differentiable at a point of ?
Let be the map given by
Prove that is differentiable at all points with .
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4.II.13F
2004 commentState the inverse function theorem for maps , where is a non-empty open subset of .
Let be the function defined by
Find a non-empty open subset of such that is locally invertible on , and compute the derivative of the local inverse.
Let be the set of all points in satisfying
Prove that is locally invertible at all points of except and . Deduce that, for each point in except and , there exist open intervals containing , respectively, such that for each in , there is a unique point in with in .
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1.I.5A
2004 commentDetermine the poles of the following functions and calculate their residues there. (i) , (ii) , (iii) .
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1.II.16A
2004 commentLet and be two polynomials such that
where are distinct non-real complex numbers and . Using contour integration, determine
carefully justifying all steps.
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2.I.5A
2004 commentLet the functions and be analytic in an open, nonempty domain and assume that there. Prove that if in then there exists such that .
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2.II.16A
2004 commentProve by using the Cauchy theorem that if is analytic in the open disc then there exists a function , analytic in , such that , .
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4.I.5A
2004 commentState and prove the Parseval formula.
[You may use without proof properties of convolution, as long as they are precisely stated.]
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4.II.15A
2004 comment(i) Show that the inverse Fourier transform of the function
is
(ii) Determine, by using Fourier transforms, the solution of the Laplace equation
given in the strip , together with the boundary conditions
where has been given above.
[You may use without proof properties of Fourier transforms.]
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1.I.9C
2004 commentFrom the general mass-conservation equation, show that the velocity field of an incompressible fluid is solenoidal, i.e. that .
Verify that the two-dimensional flow
is solenoidal and find a streamfunction such that .
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1.II.20C
2004 commentA layer of water of depth flows along a wide channel with uniform velocity , in Cartesian coordinates , with measured downstream. The bottom of the channel is at , and the free surface of the water is at . Waves are generated on the free surface so that it has the new position .
Write down the equation and the full nonlinear boundary conditions for the velocity potential (for the perturbation velocity) and the motion of the free surface.
By linearizing these equations about the state of uniform flow, show that
where is the acceleration due to gravity.
Hence, determine the dispersion relation for small-amplitude surface waves
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3.I.10C
2004 commentState Bernoulli's equation for unsteady motion of an irrotational, incompressible, inviscid fluid subject to a conservative body force .
A long vertical U-tube of uniform cross section contains an inviscid, incompressible fluid whose surface, in equilibrium, is at height above the base. Derive the equation
governing the displacement of the surface on one side of the U-tube, where is time and is the acceleration due to gravity.
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3.II.21C
2004 commentUse separation of variables to determine the irrotational, incompressible flow
around a solid sphere of radius translating at velocity along the direction in spherical polar coordinates and .
Show that the total kinetic energy of the fluid is
where is the mass of fluid displaced by the sphere.
A heavy sphere of mass is released from rest in an inviscid fluid. Determine its speed after it has fallen through a distance in terms of and .
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4.I.8C
2004 commentWrite down the vorticity equation for the unsteady flow of an incompressible, inviscid fluid with no body forces acting.
Show that the flow field
has uniform vorticity of magnitude for some constant .
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4.II.18C
2004 commentUse Euler's equation to derive the momentum integral
for the steady flow and pressure of an inviscid,incompressible fluid of density , where is a closed surface with normal .
A cylindrical jet of water of area and speed impinges axisymmetrically on a stationary sphere of radius and is deflected into a conical sheet of vertex angle as shown. Gravity is being ignored.

Use a suitable form of Bernoulli's equation to determine the speed of the water in the conical sheet, being careful to state how the equation is being applied.
Use conservation of mass to show that the width of the sheet far from the point of impact is given by
where is the distance along the sheet measured from the vertex of the cone.
Finally, use the momentum integral to determine the net force on the sphere in terms of and .
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2004
commentLet be the topology on consisting of the empty set and all sets such that is finite. Let be the usual topology on , and let be the topology on consisting of the empty set and all sets of the form for some real .
(i) Prove that all continuous functions are constant.
(ii) Give an example with proof of a non-constant function that is continuous.
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2.II.15E
2004 comment(i) Let be the set of all infinite sequences such that for all . Let be the collection of all subsets such that, for every there exists such that whenever . Prove that is a topology on .
(ii) Let a distance be defined on by
Prove that is a metric and that the topology arising from is the same as .
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3.I.5E
2004 commentLet be the contour that goes once round the boundary of the square
in an anticlockwise direction. What is ? Briefly justify your answer.
Explain why the integrals along each of the four edges of the square are equal.
Deduce that .
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2004
comment(i) Explain why the formula
defines a function that is analytic on the domain . [You need not give full details, but should indicate what results are used.]
Show also that for every such that is defined.
(ii) Write for whenever with and . Let be defined by the formula
Prove that is analytic on .
[Hint: What would be the effect of redefining to be when , and ?]
(iii) Determine the nature of the singularity of at .
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4.I.4E
2004 comment(i) Let be the open unit disc of radius 1 about the point . Prove that there is an analytic function such that for every .
(ii) Let , Re . Explain briefly why there is at most one extension of to a function that is analytic on .
(iii) Deduce that cannot be extended to an analytic function on .
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4.II.14E
2004 comment(i) State and prove Rouché's theorem.
[You may assume the principle of the argument.]
(ii) Let . Prove that the polynomial has three roots with modulus less than 3. Prove that one root satisfies ; another, , satisfies , Im ; and the third, , has .
(iii) For sufficiently small , prove that .
[You may use results from the course if you state them precisely.]
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1.I.3G
2004 commentUsing the Riemannian metric
define the length of a curve and the area of a region in the upper half-plane .
Find the hyperbolic area of the region .
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1.II.14G
2004 commentShow that for every hyperbolic line in the hyperbolic plane there is an isometry of which is the identity on but not on all of . Call it the reflection .
Show that every isometry of is a composition of reflections.
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3.I.3G
2004 commentState Euler's formula for a convex polyhedron with faces, edges, and vertices.
Show that any regular polyhedron whose faces are pentagons has the same number of vertices, edges and faces as the dodecahedron.
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3.II.15G
2004 commentLet be the lengths of a right-angled triangle in spherical geometry, where is the hypotenuse. Prove the Pythagorean theorem for spherical geometry in the form
Now consider such a spherical triangle with the sides replaced by for a positive number . Show that the above formula approaches the usual Pythagorean theorem as approaches zero.
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1.I.6B
2004 commentWrite down the general isotropic tensors of rank 2 and 3 .
According to a theory of magnetostriction, the mechanical stress described by a second-rank symmetric tensor is induced by the magnetic field vector . The stress is linear in the magnetic field,
where is a third-rank tensor which depends only on the material. Show that can be non-zero only in anisotropic materials.
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1.II.17B
2004 commentThe equation governing small amplitude waves on a string can be written as
The end points and are fixed at . At , the string is held stationary in the waveform,
The string is then released. Find in the subsequent motion.
Given that the energy
is constant in time, show that
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2.I.6B
2004 commentWrite down the general form of the solution in polar coordinates to Laplace's equation in two dimensions.
Solve Laplace's equation for in and in , subject to the conditions
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2.II.17B
2004 commentLet be the moment-of-inertia tensor of a rigid body relative to the point . If is the centre of mass of the body and the vector has components , show that
where is the mass of the body.
Consider a cube of uniform density and side , with centre at the origin. Find the inertia tensor about the centre of mass, and thence about the corner .
Find the eigenvectors and eigenvalues of .
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3.I.6D
2004 commentLet
For any variation with , show that when with
By using integration by parts, show that
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3.II.18D
2004 commentStarting from the Euler-Lagrange equations, show that the condition for the variation of the integral to be stationary is
In a medium with speed of light the ray path taken by a light signal between two points satisfies the condition that the time taken is stationary. Consider the region and suppose . Derive the equation for the light ray path . Obtain the solution of this equation and show that the light ray between and is given by
if .
Sketch the path for close to and evaluate the time taken for a light signal between these points.
[The substitution , for some constant , should prove useful in solving the differential equation.]
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4.I.6C
2004 commentChebyshev polynomials satisfy the differential equation
where is an integer.
Recast this equation into Sturm-Liouville form and hence write down the orthogonality relationship between and for .
By writing , or otherwise, show that the polynomial solutions of ( ) are proportional to .
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4.II.16C
2004 commentObtain the Green function satisfying
where is real, subject to the boundary conditions
[Hint: You may find the substitution helpful.]
Use the Green function to determine that the solution of the differential equation
subject to the boundary conditions
is
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2.I.9A
2004 commentDetermine the coefficients of Gaussian quadrature for the evaluation of the integral
that uses two function evaluations.
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2.II.20A
2004 commentGiven an matrix and , prove that the vector is the solution of the least-squares problem for if and only if . Let
Determine the solution of the least-squares problem for .
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3.I.11A
2004 commentThe linear system
where real and are given, is solved by the iterative procedure
Determine the conditions on that guarantee convergence.
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3.II.22A
2004 commentGiven , we approximate by the linear combination
By finding the Peano kernel, determine the least constant such that
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3.I.12G
2004 commentConsider the two-person zero-sum game Rock, Scissors, Paper. That is, a player gets 1 point by playing Rock when the other player chooses Scissors, or by playing Scissors against Paper, or Paper against Rock; the losing player gets point. Zero points are received if both players make the same move.
Suppose player one chooses Rock and Scissors (but never Paper) with probabilities and . Write down the maximization problem for player two's optimal strategy. Determine the optimal strategy for each value of .
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3.II.23G
2004 commentConsider the following linear programming problem:
Write down the Phase One problem in this case, and solve it.
By using the solution of the Phase One problem as an initial basic feasible solution for the Phase Two simplex algorithm, solve the above maximization problem. That is, find the optimal tableau and read the optimal solution and optimal value from it.
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4.I.10G
2004 commentState and prove the max flow/min cut theorem. In your answer you should define clearly the following terms: flow; maximal flow; cut; capacity.
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4.II.20G
2004 commentFor any number , find the minimum and maximum values of
subject to . Find all the points at which the minimum and maximum are attained. Justify your answer.
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1.I.8D
2004 commentFrom the time-dependent Schrödinger equation for , derive the equation
for and some suitable .
Show that is a solution of the time-dependent Schrödinger equation with zero potential for suitable and calculate and . What is the interpretation of this solution?
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1.II.19D
2004 commentThe angular momentum operators are . Write down their commutation relations and show that . Let
and show that
Verify that , where , for any function . Show that
for any integer . Show that is an eigenfunction of and determine its eigenvalue. Why must be an eigenfunction of ? What is its eigenvalue?
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2.I.8D
2004 commentA quantum mechanical system is described by vectors . The energy eigenvectors are
with energies respectively. The system is in the state at time . What is the probability of finding it in the state at a later time
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2.II.19D
2004 commentConsider a Hamiltonian of the form
where is a real function. Show that this can be written in the form , for some real to be determined. Show that there is a wave function , satisfying a first-order equation, such that . If is a polynomial of degree , show that must be odd in order for to be normalisable. By considering show that all energy eigenvalues other than that for must be positive.
For , use these results to find the lowest energy and corresponding wave function for the harmonic oscillator Hamiltonian
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3.I
2004 commentWrite down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum is quantised so that
for integer . Assuming circular orbits, show that the radius of the 'th orbit is
and determine . Show that the corresponding energy is then
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3.II.20D
2004 commentA one-dimensional system has the potential
For energy , the wave function has the form
By considering the relation between incoming and outgoing waves explain why we should expect
Find four linear relations between . Eliminate and show that
where and . By using the result for , or otherwise, explain why the solution for is
For define the transmission coefficient and show that, for large ,
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3.I
2004 commentWrite down the Lorentz transformation with one space dimension between two inertial frames and moving relatively to one another at speed .
A particle moves at velocity in frame . Find its velocity in frame and show that is always less than .
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4.I.7D
2004 commentFor a particle with energy and momentum , explain why an observer moving in the -direction with velocity would find
where . What is the relation between and for a photon? Show that the same relation holds for and and that
What happens for ?
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4.II.17D
2004 commentState how the 4 -momentum of a particle is related to its energy and 3momentum. How is related to the particle mass? For two particles with 4 -momenta and find a Lorentz-invariant expression that gives the total energy in their centre of mass frame.
A photon strikes an electron at rest. What is the minimum energy it must have in order for it to create an electron and positron, of the same mass as the electron, in addition to the original electron? Express the result in units of .
[It may be helpful to consider the minimum necessary energy in the centre of mass frame.]
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1.I.10H
2004 commentUse the generalized likelihood-ratio test to derive Student's -test for the equality of the means of two populations. You should explain carefully the assumptions underlying the test.
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1.II.21H
2004 commentState and prove the Rao-Blackwell Theorem.
Suppose that are independent, identically-distributed random variables with distribution
where , is an unknown parameter. Determine a one-dimensional sufficient statistic, , for .
By first finding a simple unbiased estimate for , or otherwise, determine an unbiased estimate for which is a function of .
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2.I.10H
2004 commentA study of 60 men and 90 women classified each individual according to eye colour to produce the figures below.
\begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & Blue & Brown & Green \ \hline Men & 20 & 20 & 20 \ \hline Women & 20 & 50 & 20 \ \hline \end{tabular}
Explain how you would analyse these results. You should indicate carefully any underlying assumptions that you are making.
A further study took 150 individuals and classified them both by eye colour and by whether they were left or right handed to produce the following table.
\begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & Blue & Brown & Green \ \hline Left Handed & 20 & 20 & 20 \ \hline Right Handed & 20 & 50 & 20 \ \hline \end{tabular}
How would your analysis change? You should again set out your underlying assumptions carefully.
[You may wish to note the following percentiles of the distribution.
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2.II.21H
2004 commentDefining carefully the terminology that you use, state and prove the NeymanPearson Lemma.
Let be a single observation from the distribution with density function
for an unknown real parameter . Find the best test of size , of the hypothesis against , where .
When , for which values of and will the power of the best test be at least ?
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4.I
2004 commentSuppose that are independent random variables, with having the normal distribution with mean and variance ; here are unknown and are known constants.
Derive the least-squares estimate of .
Explain carefully how to test the hypothesis against .
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4.II.19H
2004 commentIt is required to estimate the unknown parameter after observing , a single random variable with probability density function ; the parameter has the prior distribution with density and the loss function is . Show that the optimal Bayesian point estimate minimizes the posterior expected loss.
Suppose now that and , where is known. Determine the posterior distribution of given .
Determine the optimal Bayesian point estimate of in the cases when
(i) , and
(ii) .
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1.I.1H
2004 commentSuppose that is a linearly independent set of distinct elements of a vector space and spans . Prove that may be reordered, as necessary, so that spans .
Suppose that is a linearly independent set of distinct elements of and that spans . Show that .
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1.II.12H
2004 commentLet and be subspaces of the finite-dimensional vector space . Prove that both the sum and the intersection are subspaces of . Prove further that
Let be the kernels of the maps given by the matrices and respectively, where
Find a basis for the intersection , and extend this first to a basis of , and then to a basis of .
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2.I.1E
2004 commentFor each let be the matrix defined by
What is Justify your answer.
[It may be helpful to look at the cases before tackling the general case.]
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2.II.12E
2004 commentLet be a quadratic form on a real vector space of dimension . Prove that there is a basis with respect to which is given by the formula
Prove that the numbers and are uniquely determined by the form . By means of an example, show that the subspaces and need not be uniquely determined by .
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3.I.1E
2004 commentLet be a finite-dimensional vector space over . What is the dual space of ? Prove that the dimension of the dual space is the same as that of .
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3.II.13E
2004 comment(i) Let be an -dimensional vector space over and let be an endomorphism. Suppose that the characteristic polynomial of is , where the are distinct and for every .
Describe all possibilities for the minimal polynomial and prove that there are no further ones.
(ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are .
(iii) Give an example of two matrices with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix with .
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4.I.1E
2004 commentLet be a real -dimensional inner-product space and let be a dimensional subspace. Let be an orthonormal basis for . In terms of this basis, give a formula for the orthogonal projection .
Let . Prove that is the closest point in to .
[You may assume that the sequence can be extended to an orthonormal basis of .]
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4.II.11E
2004 comment(i) Let be an -dimensional inner-product space over and let be a Hermitian linear map. Prove that has an orthonormal basis consisting of eigenvectors of .
(ii) Let be another Hermitian map. Prove that is Hermitian if and only if .
(iii) A Hermitian map is positive-definite if for every non-zero vector . If is a positive-definite Hermitian map, prove that there is a unique positivedefinite Hermitian map such that .
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2004
commentLet be a finite group of order . Let be a subgroup of . Define the normalizer of , and prove that the number of distinct conjugates of is equal to the index of in . If is a prime dividing , deduce that the number of Sylow -subgroups of must divide .
[You may assume the existence and conjugacy of Sylow subgroups.]
Prove that any group of order 72 must have either 1 or 4 Sylow 3-subgroups.
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1.II.13F
2004 commentState the structure theorem for finitely generated abelian groups. Prove that a finitely generated abelian group is finite if and only if there exists a prime such that .
Show that there exist abelian groups such that for all primes . Prove directly that your example of such an is not finitely generated.
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2.I.2F
2004 commentProve that the alternating group is simple.
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2.II.13F
2004 commentLet be a subgroup of a group . Prove that is normal if and only if there is a group and a homomorphism such that
Let be the group of all matrices with in and . Let be a prime number, and take to be the subset of consisting of all with and Prove that is a normal subgroup of
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2004
commentLet be the subring of all in of the form
where and are in and . Prove that is a non-negative element of , for all in . Prove that the multiplicative group of units of has order 6 . Prove that is the intersection of two prime ideals of .
[You may assume that is a unique factorization domain.]
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2004
commentLet be the group consisting of 3-dimensional row vectors with integer components. Let be the subgroup of generated by the three vectors
(i) What is the index of in ?
(ii) Prove that is not a direct summand of .
(iii) Is the subgroup generated by and a direct summand of ?
(iv) What is the structure of the quotient group ?
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4.I.2F
2004 commentState Gauss's lemma and Eisenstein's irreducibility criterion. Prove that the following polynomials are irreducible in :
(i) ;
(ii) ;
(iii) , where is any prime number.
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4.II.12F
2004 commentAnswer the following questions, fully justifying your answer in each case.
(i) Give an example of a ring in which some non-zero prime ideal is not maximal.
(ii) Prove that is not a principal ideal domain.
(iii) Does there exist a field such that the polynomial is irreducible in ?
(iv) Is the ring an integral domain?
(v) Determine all ring homomorphisms .
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1.I.7B
2004 commentWrite down Maxwell's equations and show that they imply the conservation of charge.
In a conducting medium of conductivity , where , show that any charge density decays in time exponentially at a rate to be determined.
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1.II.18B
2004 commentInside a volume there is an electrostatic charge density , which induces an electric field with associated electrostatic potential . The potential vanishes on the boundary of . The electrostatic energy is
Derive the alternative form
A capacitor consists of three identical and parallel thin metal circular plates of area positioned in the planes and , with , with centres on the axis, and at potentials and 0 respectively. Find the electrostatic energy stored, verifying that expressions (1) and (2) give the same results. Why is the energy minimal when ?
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2.I.7B
2004 commentWrite down the two Maxwell equations that govern steady magnetic fields. Show that the boundary conditions satisfied by the magnetic field on either side of a sheet carrying a surface current of density , with normal to the sheet, are
Write down the force per unit area on the surface current.
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2.II.18B
2004 commentThe vector potential due to a steady current density is given by
where you may assume that extends only over a finite region of space. Use to derive the Biot-Savart law
A circular loop of wire of radius carries a current . Take Cartesian coordinates with the origin at the centre of the loop and the -axis normal to the loop. Use the BiotSavart law to show that on the -axis the magnetic field is in the axial direction and of magnitude
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3.I.7B
2004 commentA wire is bent into the shape of three sides of a rectangle and is held fixed in the plane, with base and , and with arms and . A second wire moves smoothly along the arms: and with . The two wires have resistance per unit length and mass per unit length. There is a time-varying magnetic field in the -direction.
Using the law of induction, find the electromotive force around the circuit made by the two wires.
Using the Lorentz force, derive the equation
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3.II.19B
2004 commentStarting from Maxwell's equations, derive the law of energy conservation in the form
where and .
Evaluate and for the plane electromagnetic wave in vacuum
where the relationships between and should be determined. Show that the electromagnetic energy propagates at speed , i.e. show that .
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1.I.11H
2004 commentLet be a transition matrix. What does it mean to say that is (a) irreducible, recurrent?
Suppose that is irreducible and recurrent and that the state space contains at least two states. Define a new transition matrix by
Prove that is also irreducible and recurrent.
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1.II.22H
2004 commentConsider the Markov chain with state space and transition matrix
Determine the communicating classes of the chain, and for each class indicate whether it is open or closed.
Suppose that the chain starts in state 2 ; determine the probability that it ever reaches state 6 .
Suppose that the chain starts in state 3 ; determine the probability that it is in state 6 after exactly transitions, .
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2.I.11H
2004 commentLet be an irreducible, positive-recurrent Markov chain on the state space with transition matrix and initial distribution , where is the unique invariant distribution. What does it mean to say that the Markov chain is reversible?
Prove that the Markov chain is reversible if and only if for all .
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2.II.22H
2004 commentConsider a Markov chain on the state space with transition probabilities as illustrated in the diagram below, where and .

For each value of , determine whether the chain is transient, null recurrent or positive recurrent.
When the chain is positive recurrent, calculate the invariant distribution.
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A2.7
2004 comment(i) What is a geodesic on a surface with Riemannian metric, and what are geodesic polar co-ordinates centred at a point on ? State, without proof, formulae for the Riemannian metric and the Gaussian curvature in terms of geodesic polar co-ordinates.
(ii) Show that a surface with constant Gaussian curvature 0 is locally isometric to the Euclidean plane.
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A3.7
2004 comment(i) The catenoid is the surface in Euclidean , with co-ordinates and Riemannian metric obtained by rotating the curve about the -axis, while the helicoid is the surface swept out by a line which lies along the -axis at time , and at time is perpendicular to the -axis, passes through the point and makes an angle with the -axis.
Find co-ordinates on each of and and write in terms of these co-ordinates.
(ii) Compute the induced Riemannian metrics on and in terms of suitable coordinates. Show that and are locally isometric. By considering the -axis in , show that this local isometry cannot be extended to a rigid motion of any open subset of Euclidean .
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A4.7
2004 commentWrite an essay on the Gauss-Bonnet theorem and its proof.
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A1.8
2004 comment(i) Let be a connected graph of order such that for any two vertices and ,
Show that if then has a path of length , and if then is Hamiltonian.
(ii) State and prove Hall's theorem.
[If you use any form of Menger's theorem, you must state it clearly.]
Let be a graph with directed edges. For , let
Find a necessary and sufficient condition, in terms of the sizes of the sets , for the existence of a set such that at every vertex there is exactly one incoming edge and exactly one outgoing edge belonging to .
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A2.8
2004 comment(i) State a result of Euler, relating the number of vertices, edges and faces of a plane graph. Show that if is a plane graph then .
(ii) Define the chromatic polynomial of a graph . Show that
where are non-negative integers. Explain, with proof, how the chromatic polynomial is related to the number of vertices, edges and triangles in . Show that if is a cycle of length , then
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A4.9
2004 commentWrite an essay on trees. You should include a proof of Cayley's result on the number of labelled trees of order .
Let be a graph of order . Which of the following statements are equivalent to the statement that is a tree? Give a proof or counterexample in each case.
(a) is acyclic and .
(b) is connected and .
(c) is connected, triangle-free and has at least two leaves.
(d) has the same degree sequence as , for some tree .
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A1.9
2004 comment(i) State the law of quadratic reciprocity. For an odd prime, evaluate the Legendre symbol
(ii) (a) Let and be distinct odd primes. Show that there exists an integer that is a quadratic residue modulo each of and a quadratic non-residue modulo each of .
(b) Let be an odd prime. Show that
(c) Let be an odd prime. Using (b) or otherwise, evaluate
Hint for : Use the equality , valid when does not divide
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A3.9
2004 comment(i) Find a solution in integers of the Pell equation .
(ii) Define the continued fraction expansion of a real number and show that it converges to .
Show that if is a nonsquare integer and and are integer solutions of , then is a convergent of .
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A4.10
2004 commentWrite an essay on pseudoprimes and their role in primality testing. You should discuss pseudoprimes, Carmichael numbers, and Euler and strong pseudoprimes. Where appropriate, your essay should include small examples to illustrate your statements.
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A1.10
2004 comment(i) What is a linear code? What does it mean to say that a linear code has length and minimum weight ? When is a linear code perfect? Show that, if , there exists a perfect linear code of length and minimum weight 3 .
(ii) Describe the construction of a Reed-Muller code. Establish its information rate and minimum weight.
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A2.9
2004 comment(i) Describe how a stream cypher operates. What is a one-time pad?
A one-time pad is used to send the message which is encoded as 0101011. By mistake, it is reused to send the message which is encoded as 0100010. Show that is one of two possible messages, and find the two possibilities.
(ii) Describe the RSA system associated with a public key , a private key and the product of two large primes.
Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack. [You are not asked to give an explicit signature system.]
Explain how to factorise when and are known.
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A2.10
2004 comment(i) Define the minimum path and the maximum tension problems for a network with span intervals specified for each arc. State without proof the connection between the two problems, and describe the Max Tension Min Path algorithm of solving them.
(ii) Find the minimum path between nodes and in the network below. The span intervals are displayed alongside the arcs.

Part II 2004
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A3.10
2004 comment(i) Consider the problem
where and . State the Lagrange Sufficiency Theorem for problem . What is meant by saying that this problem is strong Lagrangian? How is this related to the Lagrange Sufficiency Theorem? Define a supporting hyperplane and state a condition guaranteeing that problem is strong Lagrangian.
(ii) Define the terms flow, divergence, circulation, potential and differential for a network with nodes and .
State the feasible differential problem for a network with span intervals
State, without proof, the Feasible Differential Theorem.
[You must carefully define all quantities used in your statements.]
Show that the network below does not support a feasible differential.

Part II 2004
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A4.11
2004 comment(i) Consider an unrestricted geometric programming problem
where is given by
with and positive coefficients . State the dual problem of and show that if is a dual optimum then any positive solution to the system
gives an optimum for primal problem . Here is the dual objective function.
(ii) An amount of ore has to be moved from a pit in an open rectangular skip which is to be ordered from a supplier.
The skip cost is per for the bottom and two side walls and per for the front and the back walls. The cost of loading ore into the skip is per , the cost of lifting is per , and the cost of unloading is per . The cost of moving an empty skip is negligible.
Write down an unconstrained geometric programming problem for the optimal size (length, width, height) of skip minimizing the cost of moving of ore. By considering the dual problem, or otherwise, find the optimal cost and the optimal size of the skip.
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A1.13
2004 comment(i) Assume that the -dimensional vector may be written as , where is a given matrix of is an unknown vector, and
Let . Find , the least-squares estimator of , and state without proof the joint distribution of and .
(ii) Now suppose that we have observations and consider the model
where are fixed parameters with , and may be assumed independent normal variables, with , where is unknown.
(a) Find , the least-squares estimators of .
(b) Find the least-squares estimators of under the hypothesis for all .
(c) Quoting any general theorems required, explain carefully how to test , assuming is true.
(d) What would be the effect of fitting the model , where now are all fixed unknown parameters, and has the distribution given above?
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A2.12
2004 comment(i) Suppose we have independent observations , and we assume that for is Poisson with mean , and , where are given covariate vectors each of dimension , where is an unknown vector of dimension , and . Assuming that span , find the equation for , the maximum likelihood estimator of , and write down the large-sample distribution of .
(ii) A long-term agricultural experiment had 90 grassland plots, each , differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH, which for the given study has possible values "low", "medium" or "high", each taken 30 times. Explain the commands input, and interpret the resulting output in the (slightly edited) output below, in which "species" represents the species count.
(The first and last 2 lines of the data are reproduced here as an aid. You may assume that the factor pH has been correctly set up.)


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A4.14
2004 commentSuppose that are independent observations, with having probability density function of the following form
where and . You should assume that is a known function, and are unknown parameters, with , and also are given linearly independent covariate vectors. Show that
where is the log-likelihood and .
Discuss carefully the (slightly edited) output given below, and briefly suggest another possible method of analysis using the function ( ).
1:
7:
Read 6 items
1: 327172565065248688773520
Read 6 items
gender <-
1: b b b g g g
Read 6 items
age <-
1: 13&under 14-18 19&over
4: 13&under 14-18 19&over
7 :
Read 6 items
gender <- factor (gender) ; age <- factor (age)
gender age, binomial, weights
Coefficients:

Null deviance: on 5 degrees of freedom
Residual deviance: on 2 degrees of freedom
Number of Fisher Scoring iterations: 3
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A1.14
2004 comment(i) Each particle in a system of identical fermions has a set of energy levels with degeneracy , where . Derive the expression
for the mean number of particles with energy . Explain the physical significance of the parameters and .
(ii) The spatial eigenfunctions of energy for an electron of mass moving in two dimensions and confined to a square box of side are
where . Calculate the associated energies.
Hence show that when is large the number of states in energy range is
How is this formula modified when electron spin is taken into account?
The box is filled with electrons in equilibrium at temperature . Show that the chemical potential is given by
where is the number of particles per unit area in the box.
What is the value of in the limit ?
Calculate the total energy of the lowest state of the system of particles as a function of and .
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A2.14
2004 comment(i) A simple model of a crystal consists of an infinite linear array of sites equally spaced with separation . The probability amplitude for an electron to be at the -th site is . The Schrödinger equation for the is
where is real and positive. Show that the allowed energies of the electron must lie in a band , and that the dispersion relation for written in terms of a certain parameter is given by
What is the physical interpretation of and ?
(ii) Explain briefly the idea of group velocity and show that it is given by
for an electron of momentum and energy .
An electron of charge confined to one dimension moves in a periodic potential under the influence of an electric field . Show that the equation of motion for the electron is
where is the group velocity of the electron at time . Explain why
can be interpreted as an effective mass.
Show briefly how the absence from a band of an electron of charge and effective mass can be interpreted as the presence of a 'hole' carrier of charge and effective mass .
In the model of Part (i) show that
(a) for an electron behaves like a free particle of mass ;
(b) for a hole behaves like a free particle of mass .
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A4.16
2004 commentExplain the operation of the junction. Your account should include a discussion of the following topics:
(a) the rôle of doping and the fermi-energy;
(b) the rôle of majority and minority carriers;
(c) the contact potential;
(d) the relationship between the current flowing through the junction and the external voltage applied across the junction;
(e) the property of rectification.
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A1.16
2004 comment(i) Consider a homogeneous and isotropic universe with mass density , pressure and scale factor . As the universe expands its energy decreases according to the thermodynamic relation where is the volume. Deduce the fluid conservation law
Apply the conservation of total energy (kinetic plus gravitational potential) to a test particle on the edge of a spherical region in this universe to obtain the Friedmann equation
where is a constant. State clearly any assumptions you have made.
(ii) Our universe is believed to be flat and filled with two major components: pressure-free matter and dark energy with equation of state where the mass densities today are given respectively by and . Assume that each component independently satisfies the fluid conservation equation to show that the total mass density can be expressed as
where we have set .
Now consider the substitution in the Friedmann equation to show that the solution for the scale factor can be written in the form
where and are constants. Setting , specify and in terms of and . Show that the scale factor has the expected behaviour for an Einstein-de Sitter universe at early times and that the universe accelerates at late times .
[Hint: Recall that .]
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A3.14
2004 comment(i) In equilibrium, the number density of a non-relativistic particle species is given by
where is the mass, is the chemical potential and is the spin degeneracy. At around seconds, deuterium forms through the nuclear fusion of nonrelativistic protons and neutrons via the interaction:
What is the relationship between the chemical potentials of the three species when they are in chemical equilibrium? Show that the ratio of their number densities can be expressed as
where the deuterium binding energy is and you may take . Now consider the fractional densities , where is the baryon number of the universe, to re-express the ratio above in the form
which incorporates the baryon-to-photon ratio of the universe. [You may assume that the photon density is .] From this expression, explain why deuterium does not form until well below the temperature .
(ii) The number density for a photon gas in equilibrium is given by the formula
where is the photon frequency. By considering the substitution , show that the photon number density can be expressed in the form
where the constant need not be evaluated explicitly.
State the equation of state for a photon gas and explain why the chemical potential of the photon vanishes. Assuming that the photon energy density , use the first law to show that the entropy density is given by
Hence explain why, when photons are in equilibrium at early times in our universe, their temperature varies inversely with the scale factor: .
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A4.18
2004 comment(a) Consider an ideal gas of Fermi particles obeying the Pauli exclusion principle with a set of one-particle energy eigenstates . Given the probability at temperature that there are particles in the eigenstate :
determine the appropriate normalization factor . Use this to find the average number of Fermi particles in the eigenstate .
Explain briefly why in generalizing these discrete eigenstates to a continuum in momentum space (in the range to ) we must multiply by the density of states
where is the degeneracy of the eigenstates and is the volume.
(b) With the energy expressed as a momentum integral
consider the effect of changing the volume so slowly that the occupation numbers do not change (i.e. particle number and entropy remain fixed). Show that the momentum varies as and so deduce from the first law expression
that the pressure is given by
Show that in the non-relativistic limit where is the internal energy, while for ultrarelativistic particles .
(c) Now consider a Fermi gas in the limit with all momentum eigenstates filled up to the Fermi momentum . Explain why the number density can be written as
From similar expressions for the energy, deduce in both the non-relativistic and ultrarelativistic limits that the pressure may be written as
where should be specified in each case.
(d) Examine the stability of an object of radius consisting of such a Fermi degenerate gas by comparing the gravitational binding energy with the total kinetic energy. Briefly point out the relevance of these results to white dwarfs and neutron stars.
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A1.19
2004 comment(i) State and prove Maschke's theorem for finite-dimensional representations of finite groups.
(ii) is the group of bijections on . Find the irreducible representations of , state their dimensions and give their character table.
Let be the set of objects . The operation of the permutation group on is defined by the operation of the elements of separately on each index and . For example,
By considering a representative operator from each conjugacy class of , find the table of group characters for the representation of acting on . Hence, deduce the irreducible representations into which decomposes.
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A3.15
2004 comment(i) Show that the character of an transformation in the dimensional irreducible representation is given by
What are the characters of irreducible representations?
(ii) The isospin representation of two-particle states of pions and nucleons is spanned by the basis .
Pions form an isospin triplet with ; and nucleons form an isospin doublet with . Find the values of the isospin for the irreducible representations into which will decompose.
Using , write the states of the basis in terms of isospin states.
Consider the transitions
and show that their amplitudes satisfy a linear relation.
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A1.18
2004 comment(i) In an experiment, a finite amount of marker gas of diffusivity is released at time into an infinite tube in the neighbourhood of the origin . Starting from the one-dimensional diffusion equation for the concentration of marker gas,
use dimensional analysis to show that
for some dimensionless function of the similarity variable .
Write down the equation and boundary conditions satisfied by .
(ii) Consider the experiment of Part (i). Find and sketch your answer in the form of a plot of against at a few different times .
Calculate for a second experiment in which the concentration of marker gas at is instead raised to the value at and maintained at that value thereafter. Show that the total amount of marker gas released in this case becomes greater than after a time
Show further that, at much larger times than this, the concentration in the first experiment still remains greater than that in the second experiment for positions with .
[Hint: as
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A3.16
2004 comment(i) Viscous, incompressible fluid of viscosity flows steadily in the -direction in a uniform channel . The plane is fixed and the plane has constant -velocity . Neglecting gravity, derive from first principles the equations of motion of the fluid and show that the -component of the fluid velocity is and satisfies
where is the pressure in the fluid. Write down the boundary conditions on . Hence show that the volume flow rate is given by
(ii) A heavy rectangular body of width and infinite length (in the -direction) is pivoted about one edge at above a fixed rigid horizontal plane . The body has weight per unit length in the -direction, its centre of mass is distance from the pivot, and it is falling under gravity towards the fixed plane through a viscous, incompressible fluid. Let be the angle between the body and the plane. Explain the approximations of lubrication theory which permit equations (1) and (2) of Part (i) to apply to the flow in the gap between the two surfaces.
Deduce that, in the gap,
where . By taking moments about , deduce that is given by
where .
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A4.19
2004 comment(a) Solute diffuses and is advected in a moving fluid. Derive the transport equation and deduce that the solute concentration satisfies the advection-diffusion equation
where is the velocity field and the diffusivity. Write down the form this equation takes when , both and are unidirectional, in the -direction, and is a constant.
(b) A solution occupies the region , bounded by a semi-permeable membrane at across which fluid passes (by osmosis) with velocity
where is a positive constant, is a fixed uniform solute concentration in the region , and is the solute concentration in the fluid. The membrane does not allow solute to pass across , and the concentration at is a fixed value (where .
Write down the differential equation and boundary conditions to be satisfied by in a steady state. Make the equations non-dimensional by using the substitutions
and show that the concentration distribution is given by
where and should be defined, and is given by the transcendental equation
What is the dimensional fluid velocity , in terms of
(c) Show that if, instead of taking a finite value of , you had tried to take infinite, then you would have been unable to solve for unless , but in that case there would be no way of determining .
(d) Find asymptotic expansions for from equation in the following limits:
(i) For fixed, expand as a power series in , and equate coefficients to show that
(ii) For fixed, take logarithms, expand as a power series in , and show that
What is the limiting value of in the limits (i) and (ii)?
(e) Both the expansions in (d) break down when . To investigate the double limit , show that can be written as
where and is to be determined. Show that for , and for .
Briefly discuss the implication of your results for the problem raised in (c) above.
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A1.17
2004 comment(i) What is the polarisation and slowness of the time-harmonic plane elastic wave
Use the equation of motion for an isotropic homogenous elastic medium,
to show that takes one of two values and obtain the corresponding conditions on . If is complex show that .
(ii) A homogeneous elastic layer of uniform thickness -wave speed and shear modulus has a stress-free surface and overlies a lower layer of infinite depth, -wave speed and shear modulus . Show that the horizontal phase speed of trapped Love waves satisfies . Show further that
where is the horizontal wavenumber.
Assuming that (1) can be solved to give , explain how to obtain the propagation speed of a pulse of Love waves with wavenumber .
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A2.16
2004 comment(i) Sketch the rays in a small region near the relevant boundary produced by reflection and refraction of a -wave incident (a) from the mantle on the core-mantle boundary, (b) from the outer core on the inner-core boundary, and (c) from the mantle on the Earth's surface. [In each case, the region should be sufficiently small that the boundary appears to be planar.]
Describe the ray paths denoted by and .
Sketch the travel-time curves for and paths from a surface source.
(ii) From the surface of a flat Earth, an explosive source emits -waves downwards into a stratified sequence of homogeneous horizontal elastic layers of thicknesses and -wave speeds . A line of seismometers on the surface records the travel times of the various arrivals as a function of the distance from the source. Calculate the travel times, and , of the direct wave and the wave that reflects exactly once at the bottom of layer 1 .
Show that the travel time for the head wave that refracts in layer is given by
Sketch the travel-time curves for and on a single diagram and show that is tangent to .
Explain how the and can be constructed from the travel times of first arrivals provided that each head wave is the first arrival for some range of .
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A4.20
2004 commentIn a reference frame rotating about a vertical axis with angular velocity , the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible, fluid of uniform density are
where and are independent of the vertical coordinate , and is given by hydrostatic balance. State the nonlinear equations for conservation of mass and of potential vorticity for such a flow in a layer occupying . Find the pressure .
By linearising the equations about a state of rest and uniform thickness , show that small disturbances , where , to the height of the free surface obey
where and are the values of and the vorticity at .
Obtain the dispersion relation for homogeneous solutions of the form ] and calculate the group velocity of these Poincaré waves. Comment on the form of these results when and , where the lengthscale should be identified.
Explain what is meant by geostrophic balance. Find the long-time geostrophically balanced solution, and , that results from initial conditions and . Explain briefly, without detailed calculation, how the evolution from the initial conditions to geostrophic balance could be found.
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A2.17
2004 comment(i) Consider the integral equation
for in the interval , where is a real parameter and is given. Describe the method of successive approximations for solving ( ).
Suppose that
By using the Cauchy-Schwarz inequality, or otherwise, show that the successive-approximation series for converges absolutely provided
(ii) The real function satisfies the differential equation
where is a given smooth function on , subject to the boundary conditions
By integrating , or otherwise, show that obeys
Hence, or otherwise, deduce that obeys an equation of the form ( ), with
Deduce that the series solution for converges provided .
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A3.17
2004 comment(i) Give a brief description of the method of matched asymptotic expansions, as applied to a differential equation of the type
where is a non-zero constant, is a suitable smooth function and the boundary values are specified. An outline of Van Dyke's asymptotic matching principle should be included.
(ii) Consider the boundary-value problem
with . Find the integrating factor for the leading-order outer problem. Hence obtain the first two terms in the outer expansion.
Rewrite the problem using an appropriate stretched inner variable. Hence obtain the first two terms of the inner exansion.
Use van Dyke's matching principle to determine all the constants. Hence show that
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A4.21
2004 commentState Watson's lemma, describing the asymptotic behaviour of the integral
as , given that has the asymptotic expansion
as , where .
Consider the integral
where and has a unique maximum in the interval at , with , such that
By using the change of variable from to , defined by
deduce an asymptotic expansion for as . Show that the leading-order term gives
The gamma function is defined for by
By means of the substitution , or otherwise, deduce that
as
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A2.18
2004 comment(i) Let satisfy the Burgers equation
where is a positive constant. Consider solutions of the form , where and is a constant, such that
with .
Show that satisfies the so-called shock condition
By using the factorisation
where is the constant of integration, express in terms of and .
(ii) According to shallow-water theory, river waves are characterised by the PDEs
where denotes the depth of the river, denotes the mean velocity, is the constant angle of inclination, and is the constant friction coefficient.
Find the characteristic velocities and the characteristic form of the equations. Find the Riemann variables and show that if then the Riemann variables vary linearly with on the characteristics.
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A3.18
2004 comment(i) Let and denote the boundary values of functions which are analytic inside and outside the unit disc centred on the origin, respectively. Let denote the boundary of this disc. Suppose that and satisfy the jump condition
where is a constant.
Find the canonical solution of the associated homogeneous Riemann-Hilbert problem. Write down the orthogonality conditions.
(ii) Consider the linear singular integral equation
where denotes the principal value integral.
Show that the associated Riemann-Hilbert problem has the jump condition defined in Part (i) above. Using this fact, find the value of the constant that allows equation to have a solution. For this particular value of find the unique solution .
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A4.23
2004 commentLet satisfy the linear integral equation
where the measure and the contour are such that exists and is unique.
Let be defined in terms of by
(a) Show that
where
(b) Show that
,
where
(c) By recalling that the equation
admits the Lax pair
write down an expression for which gives rise to the one-soliton solution of the equation. Write down an expression for and for .
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A1.1 B1.1
2004 comment(i) Give the definitions of a recurrent and a null recurrent irreducible Markov chain.
Let be a recurrent Markov chain with state space and irreducible transition matrix . Prove that the vectors , with entries and
are -invariant:
(ii) Let be the birth and death process on with the following transition probabilities:
By relating to the symmetric simple random walk on , or otherwise, prove that is a recurrent Markov chain. By considering invariant measures, or otherwise, prove that is null recurrent.
Calculate the vectors for the chain .
Finally, let and let be the number of visits to 1 before returning to 0 . Show that .
[You may use properties of the random walk or general facts about Markov chains without proof but should clearly state them.]
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A2.1
2004 comment(i) Let be a proper subset of the finite state space of an irreducible Markov chain , whose transition matrix is partitioned as
If only visits to states in are recorded, we see a -valued Markov chain ; show that its transition matrix is
(ii) Local MP Phil Anderer spends his time in London in the Commons , in his flat , in the bar or with his girlfriend . Each hour, he moves from one to another according to the transition matrix , though his wife (who knows nothing of his girlfriend) believes that his movements are governed by transition matrix :

The public only sees Phil when he is in ; calculate the transition matrix which they believe controls his movements.
Each time the public Phil moves to a new location, he phones his wife; write down the transition matrix which governs the sequence of locations from which the public Phil phones, and calculate its invariant distribution.
Phil's wife notes down the location of each of his calls, and is getting suspicious - he is not at his flat often enough. Confronted, Phil swears his fidelity and resolves to dump his troublesome transition matrix, choosing instead

Will this deal with his wife's suspicions? Explain your answer.
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A3.1 B3.1
2004 comment(i) Give the definition of the time-reversal of a discrete-time Markov chain . Define a reversible Markov chain and check that every probability distribution satisfying the detailed balance equations is invariant.
(ii) Customers arrive in a hairdresser's shop according to a Poisson process of rate . The shop has hairstylists and waiting places; each stylist is working (on a single customer) provided that there is a customer to serve, and any customer arriving when the shop is full (i.e. the numbers of customers present is ) is not admitted and never returns. Every admitted customer waits in the queue and then is served, in the first-come-first-served order (say), the service taking an exponential time of rate ; the service times of admitted customers are independent. After completing his/her service, the customer leaves the shop and never returns.
Set up a Markov chain model for the number of customers in the shop at time . Assuming , calculate the equilibrium distribution of this chain and explain why it is unique. Show that in equilibrium is time-reversible, i.e. has the same distribution as where , and .
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A4.1
2004 comment(a) Give three definitions of a continuous-time Markov chain with a given -matrix on a finite state space: (i) in terms of holding times and jump probabilities, (ii) in terms of transition probabilities over small time intervals, and (iii) in terms of finite-dimensional distributions.
(b) A flea jumps clockwise on the vertices of a triangle; the holding times are independent exponential random variables of rate one. Find the eigenvalues of the corresponding -matrix and express transition probabilities , in terms of these roots. Deduce the formulas for the sums
in terms of the functions and .
Find the limits
What is the connection between the decompositions and
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A1.2 B1.2
2004 comment(i) In Hamiltonian mechanics the action is written
Starting from Maupertius' principle , derive Hamilton's equations
Show that is a constant of the motion if . When is a constant of the motion?
(ii) Consider the action given in Part (i), evaluated on a classical path, as a function of the final coordinates and final time , with the initial coordinates and the initial time held fixed. Show that obeys
Now consider a simple harmonic oscillator with . Setting the initial time and the initial coordinate to zero, find the classical solution for and with final coordinate at time . Hence calculate , and explicitly verify (2) in this case.
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A2.2 B2.1
2004 comment(i) Consider a light rigid circular wire of radius and centre . The wire lies in a vertical plane, which rotates about the vertical axis through . At time the plane containing the wire makes an angle with a fixed vertical plane. A bead of mass is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by . The angle between the line and the downward vertical is .
Show that the Lagrangian of the system is
Calculate two independent constants of the motion, and explain their physical significance.
(ii) A dynamical system has Hamiltonian , where is a parameter. Consider an ensemble of identical systems chosen so that the number density of systems, , in the phase space element is either zero or one. Prove Liouville's Theorem, namely that the total area of phase space occupied by the ensemble is time-independent.
Now consider a single system undergoing periodic motion . Give a heuristic argument based on Liouville's Theorem to show that the area enclosed by the orbit,
is approximately conserved as the parameter is slowly varied (i.e. that is an adiabatic invariant).
Consider , with a positive integer. Show that as is slowly varied the energy of the system, , varies as
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A3.2
2004 comment(i) Explain the concept of a canonical transformation from coordinates to . Derive the transformations corresponding to generating functions and .
(ii) A particle moving in an electromagnetic field is described by the Lagrangian
where is constant
(a) Derive the equations of motion in terms of the electric and magnetic fields and .
(b) Show that and are invariant under the gauge transformation
for .
(c) Construct the Hamiltonian. Find the generating function for the canonical transformation which implements the gauge transformation (1).
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A4.2
2004 commentConsider a system of coordinates rotating with angular velocity relative to an inertial coordinate system.
Show that if a vector is changing at a rate in the inertial system, then it is changing at a rate
with respect to the rotating system.
A solid body rotates with angular velocity in the absence of external torque. Consider the rotating coordinate system aligned with the principal axes of the body.
(a) Show that in this system the motion is described by the Euler equations
, where are the components of the angular velocity in the rotating system and are the principal moments of inertia.
(b) Consider a body with three unequal moments of inertia, . Show that rotation about the 1 and 3 axes is stable to small perturbations, but rotation about the 2 axis is unstable.
(c) Use the Euler equations to show that the kinetic energy, , and the magnitude of the angular momentum, , are constants of the motion. Show further that
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A1.3
2004 comment(i) Let be a Hilbert space, and let be a non-zero closed vector subspace of . For , show that there is a unique closest point to in .
(ii) (a) Let . Show that . Show also that if and then .
(b) Deduce that .
(c) Show that the map from to is a continuous linear map, with .
(d) Show that is the projection onto along .
Now suppose that is a subspace of that is not necessarily closed. Explain why implies that is dense in
Give an example of a subspace of that is dense in but is not equal to .
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A2.3 B2.2
2004 comment(i) Prove Riesz's Lemma, that if is a normed space and is a vector subspace of such that for some we have for all with , then is dense in . [Here denotes the distance from to .]
Deduce that any normed space whose unit ball is compact is finite-dimensional. [You may assume that every finite-dimensional normed space is complete.]
Give an example of a sequence in an infinite-dimensional normed space such that for all , but has no convergent subsequence.
(ii) Let be a vector space, and let and be two norms on . What does it mean to say that and are equivalent?
Show that on a finite-dimensional vector space all norms are equivalent. Deduce that every finite-dimensional normed space is complete.
Exhibit two norms on the vector space that are not equivalent.
In addition, exhibit two norms on the vector space that are not equivalent.
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A3.3 B3.2
2004 comment(i) Let be an infinite-dimensional Hilbert space. Show that has a (countable) orthonormal basis if and only if has a countable dense subset. [You may assume familiarity with the Gram-Schmidt process.]
State and prove Bessel's inequality.
(ii) State Parseval's equation. Using this, prove that if has a countable dense subset then there is a surjective isometry from to .
Explain carefully why the functions , form an orthonormal basis for
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A4.3
2004 commentState and prove the Dominated Convergence Theorem. [You may assume the Monotone Convergence Theorem.]
Let and be real numbers, with . Prove carefully that
[Any standard results that you use should be stated precisely.]
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A1.4 B1.3
2004 comment(i) Let be a commutative ring. Define the terms prime ideal and maximal ideal, and show that if an ideal in is maximal then is also prime.
(ii) Let be a non-trivial prime ideal in the commutative ring ('non-trivial' meaning that and ). If has finite index as a subgroup of , show that is also maximal. Give an example to show that this may fail, if the assumption of finite index is omitted. Finally, show that if is a principal ideal domain, then every non-trivial prime ideal in is maximal.
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A2.4 B2.3
2004 comment(i) State Gauss' Lemma on polynomial irreducibility. State and prove Eisenstein's criterion.
(ii) Which of the following polynomials are irreducible over ? Justify your answers.
(a)
(b)
(c) with prime
[Hint: consider substituting .]
(d) with prime.
[Hint: show any factor has degree at least two, and consider powers of dividing coefficients.]
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A3.4
2004 comment(i) Let be a field and a finite normal extension of . If is a finite subgroup of order in the Galois group , show that is a normal extension of the -invariant subfield of degree and that . [You may assume the theorem of the primitive element.]
(ii) Show that the splitting field over of the polynomial is and deduce that its Galois group has order 8. Exhibit a subgroup of order 4 of the Galois group, and determine the corresponding invariant subfield.
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A4.4
2004 comment(a) Let be the maximal power of the prime dividing the order of the finite group , and let denote the number of subgroups of of order . State clearly the numerical restrictions on given by the Sylow theorems.
If and are subgroups of of orders and respectively, and their intersection has order , show the set contains elements.
(b) The finite group has 48 elements. By computing the possible values of , show that cannot be simple.
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A B
2004 comment(i) Show that the work done in assembling a localised charge distribution in a region with an associated potential is
and that this can be written as an integral over all space
where the electric field .
(ii) What is the force per unit area on an infinite plane conducting sheet with a charge density per unit area (a) if it is isolated in space and (b) if the electric field vanishes on one side of the sheet?
An infinite cylindrical capacitor consists of two concentric cylindrical conductors with radii , carrying charges per unit length respectively. Calculate the capacitance per unit length and the energy per unit length. Next determine the total force on each conductor, and calculate the rate of change of energy of the inner and outer conductors if they are moved radially inwards and outwards respectively with speed . What is the corresponding rate of change of the capacitance?
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A2.5
2004 comment(i) Write down the general solution of Poisson's equation. Derive from Maxwell's equations the Biot-Savart law for the magnetic field of a steady localised current distribution.
(ii) A plane rectangular loop with sides of length and lies in the plane and is centred on the origin. Show that when , the vector potential is given approximately by
where is the magnetic moment of the loop.
Hence show that the magnetic field at a great distance from an arbitrary small plane loop of area , situated in the -plane near the origin and carrying a current , is given by
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A3.5 B3.3
2004 comment(i) State Maxwell's equations and show that the electric field and the magnetic field can be expressed in terms of a scalar potential and a vector potential . Hence derive the inhomogeneous wave equations that are satisfied by and respectively.
(ii) The plane separates a vacuum in the half-space from a perfectly conducting medium occupying the half-space . Derive the boundary conditions on and at .
A plane electromagnetic wave with a magnetic field , travelling in the -plane at an angle to the -direction, is incident on the interface at . If the wave has frequency show that the total magnetic field is given by
where is a constant. Hence find the corresponding electric field , and obtain the surface charge density and the surface current at the interface.
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A4.5
2004 commentConsider a frame moving with velocity v relative to the laboratory frame where . The electric and magnetic fields in are and , while those measured in are and . Given that , show that
for any closed circuit and hence that .
Now consider a fluid with electrical conductivity and moving with velocity . Use Ohm's law in the moving frame to relate the current density to the electric field in the laboratory frame, and show that if remains finite in the limit then
The magnetic helicity in a volume is given by where is the vector potential. Show that if the normal components of and both vanish on the surface bounding then .
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B2.4
2004 comment(i) Define carefully what is meant by a Hopf bifurcation in a two-dimensional dynamical system. Write down the normal form for this bifurcation, correct to cubic order, and distinguish between bifurcations of supercritical and subcritical type. Describe, without detailed calculations, how a general two-dimensional system with a Hopf bifurcation at the origin can be reduced to normal form by a near-identity transformation.
(ii) A Takens-Bogdanov bifurcation of a fixed point of a two-dimensional system is characterised by a Jacobian with the canonical form
at the bifurcation point. Consider the system
Show that a near-identity transformation of the form
exists that reduces the system to the normal (canonical) form, correct up to quadratic terms,
It is known that the general form of the equations near the bifurcation point can be written (setting )
Find all the fixed points of this system, and the values of for which these fixed points have (a) steady state bifurcations and (b) Hopf bifurcations.
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B3.4
2004 comment(i) Describe the use of the stroboscopic method for obtaining approximate solutions to the second order equation
when . In particular, by writing , obtain expressions in terms of for the rate of change of and . Evaluate these expressions when .
(ii) In planetary orbit theory a crude model of an orbit subject to perturbation from a distant body is given by the equation
where are polar coordinates in the plane, and is a positive constant.
(a) Show that when all bounded orbits are closed.
(b) Now suppose , and look for almost circular orbits with , where is a constant. By writing , and by making a suitable choice of the constant , use the stroboscopic method to find equations for and . By writing and considering , or otherwise, determine and in the case . Hence describe the orbits of the system.
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A B1.12
2004 comment(i) State and prove the Knaster-Tarski Fixed-Point Theorem.
(ii) A subset of a poset is called an up-set if whenever satisfy and then also . Show that the set of up-sets of (ordered by inclusion) is a complete poset.
Let and be totally ordered sets, such that is isomorphic to an up-set in and is isomorphic to the complement of an up-set in . Prove that is isomorphic to . Indicate clearly where in your argument you have made use of the fact that and are total orders, rather than just partial orders.
[Recall that posets and are called isomorphic if there exists a bijection from to such that, for any , we have if and only if .]
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B2.11
2004 commentDefine the sets . Show that each is transitive, and explain why whenever . Prove that every set is a member of some .
Which of the following are true and which are false? Give proofs or counterexamples as appropriate. You may assume standard properties of rank.
(a) If the rank of a set is a (non-zero) limit then is infinite.
(b) If the rank of a set is a successor then is finite.
(c) If the rank of a set is countable then is countable.
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A3.8 B3.11
2004 comment(i) State and prove the Compactness Theorem for first-order predicate logic.
State and prove the Upward Löwenheim-Skolem Theorem.
[You may use the Completeness Theorem for first-order predicate logic.]
(ii) For each of the following theories, either give axioms (in the language of posets) for the theory or prove carefully that the theory is not axiomatisable.
(a) The theory of posets having no maximal element.
(b) The theory of posets having a unique maximal element.
(c) The theory of posets having infinitely many maximal elements.
(d) The theory of posets having finitely many maximal elements.
(e) The theory of countable posets having a unique maximal element.
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A4.8 B4.10
2004 commentWrite an essay on recursive functions. Your essay should include a sketch of why every computable function is recursive, and an explanation of the existence of a universal recursive function, as well as brief discussions of the Halting Problem and of the relationship between recursive sets and recursively enumerable sets.
[You may assume that every recursive function is computable. You do not need to give proofs that particular functions to do with prime-power decompositions are recursive.]
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A1.12 B1.15
2004 comment(i) What does it mean to say that a family of densities is an exponential family?
Consider the family of densities on parametrised by the positive parameters and defined by
Prove that this family is an exponential family, and identify the natural parameters and the reference measure.
(ii) Let be a sample drawn from the above distribution. Find the maximum-likelihood estimators of the parameters . Find the Fisher information matrix of the family (in terms of the natural parameters). Briefly explain the significance of the Fisher information matrix in relation to unbiased estimation. Compute the mean of and of .
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A2.11 B2.16
2004 comment(i) In the context of a decision-theoretic approach to statistics, what is a loss function? a decision rule? the risk function of a decision rule? the Bayes risk of a decision rule? the Bayes rule with respect to a given prior distribution?
Show how the Bayes rule with respect to a given prior distribution is computed.
(ii) A sample of people is to be tested for the presence of a certain condition. A single real-valued observation is made on each one; this observation comes from density if the condition is absent, and from density if the condition is present. Suppose if the person does not have the condition, otherwise, and suppose that the prior distribution for the is that they are independent with common distribution , where is known. If denotes the observation made on the person, what is the posterior distribution of the ?
Now suppose that the loss function is defined by
for action , where are positive constants. If denotes the posterior probability that given the data, prove that the Bayes rule for this prior and this loss function is to take if exceeds the threshold value , and otherwise to take .
In an attempt to control the proportion of false positives, it is proposed to use a different loss function, namely,
where . Prove that the Bayes rule is once again a threshold rule, that is, we take action if and only if , and determine as fully as you can.
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A3.12 B3.15
2004 comment(i) What is a sufficient statistic? What is a minimal sufficient statistic? Explain the terms nuisance parameter and ancillary statistic.
(ii) Let be independent random variables with common uniform( distribution, and suppose you observe , where the positive parameters are unknown. Write down the joint density of and prove that the statistic
is minimal sufficient for . Find the maximum-likelihood estimator of .
Regarding as the parameter of interest and as the nuisance parameter, is ancillary? Find the mean and variance of . Hence find an unbiased estimator of .
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A4.13 B4.15
2004 commentSuppose that is the parameter of a non-degenerate exponential family. Derive the asymptotic distribution of the maximum-likelihood estimator of based on a sample of size . [You may assume that the density is infinitely differentiable with respect to the parameter, and that differentiation with respect to the parameter commutes with integration.]
Part II 2004
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A1.11 B1.16
2004 comment(i) What does it mean to say that is a utility function? What is a utility function with constant absolute risk aversion (CARA)?
Let denote the prices at time of risky assets, and suppose that there is also a riskless zeroth asset, whose price at time 0 is 1 , and whose price at time 1 is . Suppose that has a multivariate Gaussian distribution, with mean and non-singular covariance . An agent chooses at time 0 a portfolio of holdings of the risky assets, at total cost , and at time 1 realises his gain . Given that he wishes the mean of to be equal to , find the smallest value that the variance of can be. What is the portfolio that achieves this smallest variance? Hence sketch the region in the plane of pairs that can be achieved by some choice of , and indicate the mean-variance efficient frontier.
(ii) Suppose that the agent has a CARA utility with coefficient of absolute risk aversion. What portfolio will he choose in order to maximise ? What then is the mean of ?
Regulation requires that the agent's choice of portfolio has to satisfy the valueat-risk (VaR) constraint
where and are determined by the regulatory authority. Show that this constraint has no effect on the agent's decision if . If , will this constraint necessarily affect the agent's choice of portfolio?
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A3.11 B3.16
2004 comment(i) Consider a single-period binomial model of a riskless asset (asset 0 ), worth 1 at time 0 and at time 1 , and a risky asset (asset 1 ), worth 1 at time 0 and worth at time 1 if the period was good, otherwise worth . Assuming that
show how any contingent claim to be paid at time 1 can be priced and exactly replicated. Briefly explain the significance of the condition , and indicate how the analysis of the single-period model extends to many periods.
(ii) Now suppose that , and that the risky asset is worth at time zero. Show that the time- 0 value of an American put option with strike and expiry at time is equal to 79 , and find the optimal exercise policy.
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A4.12 B4.16
2004 commentWhat is Brownian motion ? Briefly explain how Brownian motion can be considered as a limit of simple random walks. State the Reflection Principle for Brownian motion, and use it to derive the distribution of the first passage time to some level .
Suppose that , where is constant. Stating clearly any results to which you appeal, derive the distribution of the first-passage time to .
Now let , where . Find the density of .
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A2.13 B2.22
2004 comment(i) The creation and annihilation operators for a harmonic oscillator of angular frequency satisfy the commutation relation . Write down an expression for the Hamiltonian in terms of and .
There exists a unique ground state of such that . Explain how the space of eigenstates of is formed, and deduce the eigenenergies for these states. Show that
(ii) Write down the number operator of the harmonic oscillator in terms of and . Show that
The operator is defined to be
Show that commutes with . Show also that
By considering the action of on the state show that
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A3.13 B3.21
2004 comment(i) A quantum mechanical system consists of two identical non-interacting particles with associated single-particle wave functions and energies , where Show how the states for the two lowest energy levels of the system are constructed and discuss their degeneracy when the particles have (a) spin 0 , (b) spin .
(ii) The Pauli matrices are defined to be
State how the spin operators may be expressed in terms of the Pauli matrices, and show that they describe states with total angular momentum .
An electron is at rest in the presence of a magnetic field , and experiences an interaction potential . At the state of the electron is the eigenstate of with eigenvalue . Calculate the probability that at later time the electron will be measured to be in the eigenstate of with eigenvalue .
Part II 2004
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A4.15 B4.22
2004 commentThe states of the hydrogen atom are denoted by with and associated energy eigenvalue , where
A hydrogen atom is placed in a weak electric field with interaction Hamiltonian
a) Derive the necessary perturbation theory to show that to the change in the energy associated with the state is given by
The wavefunction of the ground state is
By replacing , in the denominator of by show that
b) Find a matrix whose eigenvalues are the perturbed energies to for the states and . Hence, determine these perturbed energies to in terms of the matrix elements of between these states.
[Hint:
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A1.15 B1.24
2004 comment(i) What is an affine parameter of a timelike or null geodesic? Prove that for a timelike geodesic one may take to be proper time . The metric
with represents an expanding universe. Calculate the Christoffel symbols.
(ii) Obtain the law of spatial momentum conservation for a particle of rest mass in the form
Assuming that the energy , derive an expression for in terms of and and show that the energy is not conserved but rather that it decreases with time. In particular, show that if the particle is moving extremely relativistically then the energy decreases as , and if it is moving non-relativistically then the kinetic energy, , decreases as .
Show that the frequency of a photon emitted at time will be observed at time to have frequency
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A2.15 B2.24
2004 comment(i) State and prove Birkhoff's theorem.
(ii) Derive the Schwarzschild metric and discuss its relevance to the problem of gravitational collapse and the formation of black holes.
[Hint: You may assume that the metric takes the form
and that the non-vanishing components of the Einstein tensor are given by
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A4.17 B4.25
2004 commentStarting from the Ricci identity
give an expression for the curvature tensor of the Levi-Civita connection in terms of the Christoffel symbols and their partial derivatives. Using local inertial coordinates, or otherwise, establish that
A vector field with components satisfies
Show, using equation that
and hence that
where is the Ricci tensor. Show that equation may be written as
If the metric is taken to be the Schwarzschild metric
show that is a solution of . Calculate .
Electromagnetism can be described by a vector potential and a Maxwell field tensor satisfying
The divergence of is arbitrary and we may choose . With this choice show that in a general spacetime
Hence show that in the Schwarzschild spacetime a tensor field whose only non-trivial components are , where is a constant, satisfies the field equations .
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A1.20 B1.20
2004 comment(i) Define the Backward Difference Formula (BDF) method for ordinary differential equations and derive its two-step version.
(ii) Prove that the interval belongs to the linear stability domain of the twostep BDF method.
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A2.19 B2.20
2004 comment(i) The five-point equations, which are obtained when the Poisson equation (with Dirichlet boundary conditions) is discretized in a square, are
where for all .
Formulate the Gauss-Seidel method for the above linear system and prove its convergence. In the proof you should carefully state any theorems you use. [You may use Part (ii) of this question.]
(ii) By arranging the two-dimensional arrays and into the column vectors and respectively, the linear system described in Part (i) takes the matrix form . Prove that, regardless of the ordering of the points on the grid, the matrix is symmetric and positive definite.
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A3.19 B3.20
2004 comment(i) The diffusion equation
with the initial condition , and with zero boundary conditions at and , can be solved by the method
where , and . Prove that implies convergence.
(ii) By discretizing the same equation and employing the same notation as in Part (i), determine conditions on such that the method
is stable.
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A4.22 B4.20
2004 commentWrite an essay on the method of conjugate gradients. You should define the method, list its main properties and sketch the relevant proof. You should also prove that (in exact arithmetic) the method terminates in a finite number of steps, briefly mention the connection with Krylov subspaces, and describe the approach of preconditioned conjugate gradients.
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A1.6 B1.17
2004 comment(i) State Liapunov's First Theorem and La Salle's Invariance Principle. Use these results to show that the system
has an asymptotically stable fixed point at the origin.
(ii) Define the basin of attraction of an invariant set of a dynamical system.
Consider the equations
(a) Find the fixed points of the system and determine their type.
(b) Show that the basin of attraction of the origin includes the union over of the regions
Sketch these regions for in the case .
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A2.6 B2.17
2004 comment(i) A linear system in takes the form . Explain (without detailed calculation but by giving examples) how to classify the dynamics of the system in terms of the determinant and the trace of A. Show your classification graphically, and describe the dynamics that occurs on the boundaries of the different regions on your diagram.
(ii) A nonlinear system in has the form . The Jacobian (linearization) of at the origin is non-hyperbolic, with one eigenvalue of in the left-hand half-plane. Define the centre manifold for this system, and explain (stating carefully any results you use) how the dynamics near the origin may be reduced to a one-dimensional system on the centre manifold.
A dynamical system of this type has the form
Find the coefficients for the expansion of the centre manifold correct up to and including terms of order , and write down in terms of these coefficients the equation for the dynamics on the centre manifold up to order . Using this reduced equation, give a complete set of conditions on the coefficients that guarantee that the origin is stable.
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A4.6 B4.17
2004 comment(a) Consider the map , defined on , where , , and the constant satisfies . Give, with reasons, the values of (if any) for which the map has (i) a fixed point, (ii) a cycle of least period , (iii) an aperiodic orbit. Does the map exhibit sensitive dependence on initial conditions?
Show (graphically if you wish) that if the map has an -cycle then it has an infinite number of such cycles. Is this still true if is replaced by
(b) Consider the map
where and are defined as in Part (a), and is a parameter.
Find the regions of the plane for which the map has (i) no fixed points, (ii) exactly two fixed points.
Now consider the possible existence of a 2-cycle of the map when , and suppose the elements of the cycle are with . By expanding in powers of , so that , and similarly for and , show that
Use this result to sketch the region of the plane in which 2-cycles exist. How many distinct cycles are there for each value of in this region?
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A3.6 B3.17
2004 comment(i) Consider a system in that is almost Hamiltonian:
where and . Show that if the system has a periodic orbit then , and explain how to evaluate this orbit approximately for small . Illustrate your method by means of the system
(ii) Consider the system
(a) Show that when the system is Hamiltonian, and find the Hamiltonian. Sketch the trajectories in the case . Identify the value of for which there is a homoclinic orbit.
(b) Suppose . Show that the small change in around an orbit of the Hamiltonian system can be expressed to leading order as an integral of the form
where are the extrema of the -coordinates of the orbits of the Hamiltonian system, distinguishing between the cases .
(c) Find the value of , correct to leading order in , at which the system has a homoclinic orbit.
(d) By examining the eigenvalues of the Jacobian at the origin, determine the stability of the homoclinic orbit, being careful to state clearly any standard results that you use.
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B1.5
2004 commentState and prove Menger's theorem (vertex form).
Let be a graph of connectivity and let be disjoint subsets of with . Show that there exist vertex disjoint paths from to .
The graph is said to be -linked if, for every sequence of distinct vertices, there exist paths, , that are vertex disjoint. By removing an edge from , or otherwise, show that, for , need not be -linked even if .
Prove that if and then is -linked.
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B2.5
2004 commentState and prove Sperner's lemma on antichains.
The family is said to split if, for all distinct , there exists with but . Prove that if splits then , where .
Show moreover that, if splits and no element of is in more than members of , then .
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B4.1
2004 commentWrite an essay on Ramsey's theorem. You should include the finite and infinite versions, together with some discussion of bounds in the finite case, and give at least one application.
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B1.6
2004 comment(a) Show that every irreducible complex representation of an abelian group is onedimensional.
(b) Show, by example, that the analogue of (a) fails for real representations.
(c) Let the cyclic group of order act on by cyclic permutation of the standard basis vectors. Decompose this representation explicitly into irreducibles.
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B2.6
2004 commentLet be a group with three generators and relations , and where is a prime number.
(a) Show that . Show that the conjugacy classes of are the singletons and the sets , as range from 0 to , but .
(b) Find 1-dimensional representations of .
(c) Let be a th root of unity. Show that the following defines an irreducible representation of on :
where the are the standard basis vectors of .
(d) Show that (b) and (c) cover all irreducible isomorphism classes.
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B3.5
2004 commentCompute the character table for the group of even permutations of five elements. You may wish to follow the steps below.
(a) List the conjugacy classes in and their orders.
(b) acts on by permuting the standard basis vectors. Show that splits as , where is the trivial 1-dimensional representation and is irreducible.
(c) By using the formula for the character of the symmetric square ,
decompose to produce a 5-dimensional, irreducible representation, and find its character.
(d) Show that the exterior square decomposes into two distinct irreducibles and compute their characters, to complete the character table of .
[Hint: You can save yourself some computational effort if you can explain why the automorphism of , defined by conjugation by a transposition in , must swap the two summands of .]
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B4.2
2004 commentWrite an essay on the finite-dimensional representations of , including a proof of their complete reducibility, and a description of the irreducible representations and the decomposition of their tensor products.
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B1.7
2004 commentLet be a finite extension of fields. Define the trace and norm of an element .
Assume now that the extension is Galois, with cyclic Galois group of prime order , generated by .
i) Show that .
ii) Show that is a -vector subspace of of dimension . Deduce that if , then if and only if for some . [You may assume without proof that is surjective for any finite separable extension .]
iii) Suppose that has characteristic . Deduce from (i) that every element of can be written as for some . Show also that if , then belongs to . Deduce that is the splitting field over of for some .
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B3.6
2004 commentLet be a field, and a finite subgroup of . Show that is cyclic.
Define the cyclotomic polynomials , and show from your definition that
Deduce that is a polynomial with integer coefficients.
Let be a prime with . Let , where are irreducible. Show that for each the degree of is equal to the order of in the group .
Use this to write down an irreducible polynomial of degree 10 over .
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B4.3
2004 commentLet be a finite Galois extension of fields. Explain what is meant by the Galois correspondence between subfields of containing and subgroups of . Show that if then is a normal subgroup of if and only if is normal. What is in this case?
Let be the splitting field of over . Prove that is isomorphic to the dihedral group of order 8. Hence determine all subfields of , expressing each in the form for suitable .
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B1.8
2004 commentWhat is a smooth vector bundle over a manifold ?
Assuming the existence of "bump functions", prove that every compact manifold embeds in some Euclidean space .
By choosing an inner product on , or otherwise, deduce that for any compact manifold there exists some vector bundle such that the direct sum is isomorphic to a trivial vector bundle.
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B2 7
2004 commentFor each of the following assertions, either provide a proof or give and justify a counterexample.
[You may use, without proof, your knowledge of the de Rham cohomology of surfaces.]
(a) A smooth map must have degree zero.
(b) An embedding extends to an embedding if and only if the map
is the zero map.
(c) is orientable.
(d) The surface admits the structure of a Lie group if and only if .
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B4.4
2004 commentDefine what it means for a manifold to be oriented, and define a volume form on an oriented manifold.
Prove carefully that, for a closed connected oriented manifold of dimension , .
[You may assume the existence of volume forms on an oriented manifold.]
If and are closed, connected, oriented manifolds of the same dimension, define the degree of a map .
If has degree and , can be
(i) infinite? (ii) a single point? (iii) empty?
Briefly justify your answers.
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B2.8
2004 commentLet and be finite simplicial complexes. Define the -th chain group and the boundary homomorphism . Prove that and define the homology groups of . Explain briefly how a simplicial map induces a homomorphism of homology groups.
Suppose now that consists of the proper faces of a 3-dimensional simplex. Calculate from first principles the homology groups of . If a simplicial map gives a homeomorphism of the underlying polyhedron , is the induced homology map necessarily the identity?
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B1.9
2004 commentLet , where is a root of . Prove that has degree 3 over , and admits three distinct embeddings in . Find the discriminant of and determine the ring of integers of . Factorise and into a product of prime ideals.
Using Minkowski's bound, show that has class number 1 provided all prime ideals in dividing 2 and 3 are principal. Hence prove that has class number
[You may assume that the discriminant of is .]
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B3.7
2004 commentA finite simplicial complex is the union of subcomplexes and . Describe the Mayer-Vietoris exact sequence that relates the homology groups of to those of , and . Define all the homomorphisms in the sequence, proving that they are well-defined (a proof of exactness is not required).
A surface is constructed by identifying together (by means of a homeomorphism) the boundaries of two Möbius strips and . Assuming relevant triangulations exist, determine the homology groups of .
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B4.5
2004 commentWrite down the definition of a covering space and a covering map. State and prove the path lifting property for covering spaces and state, without proof, the homotopy lifting property.
Suppose that a group is a group of homeomorphisms of a space . Prove that, under conditions to be stated, the quotient map is a covering map and that is isomorphic to . Give two examples in which this last result can be used to determine the fundamental group of a space.
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B2.9
2004 commentLet be an integer greater than 1 and let denote a primitive -th root of unity in . Let be the ring of integers of . If is a prime number with , outline the proof that
where the are distinct prime ideals of , and with the least integer such that . [Here is the Euler -function of .
Determine the factorisations of and 11 in . For each integer , prove that, in the ring of integers of , there is a unique prime ideal dividing 2 , and a unique prime ideal dividing 3 .
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B4.6
2004 commentLet be a finite extension of , and the ring of integers of . Write an essay outlining the proof that every non-zero ideal of can be written as a product of non-zero prime ideals, and that this factorisation is unique up to the order of the factors.
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B1.10
2004 commentSuppose that and are orthonormal bases of a Hilbert space and that .
(a) Show that .
(b) Show that .
is a Hilbert-Schmidt operator if for some (and hence every) orthonormal basis .
(c) Show that the set HS of Hilbert-Schmidt operators forms a linear subspace of , and that is an inner product on ; show that this inner product does not depend on the choice of the orthonormal basis .
(d) Let be the corresponding norm. Show that , and show that a Hilbert-Schmidt operator is compact.
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B3.8
2004 commentLet be a Hilbert space. An operator in is normal if . Suppose that is normal and that . Let .
(a) Suppose that is invertible and . Show that .
(b) Show that is normal, and that .
(c) Show that is normal.
(d) Show that is unitary.
(e) Show that is Hermitian.
[You may use the fact that, if is normal, the spectral radius of is equal to ]
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B4.7
2004 commentSuppose that is a bounded linear operator on an infinite-dimensional Hilbert space , and that is real and non-negative for each .
(a) Show that is Hermitian.
(b) Let . Show that
(c) Show that is an approximate eigenvalue for .
Suppose in addition that is compact and injective.
(d) Show that is an eigenvalue for , with finite-dimensional eigenspace.
Explain how this result can be used to diagonalise .
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B1.11
2004 commentLet be a fixed complex number with positive imaginary part. For , define
Prove the identities
and deduce that . Show further that is the only zero of in the parallelogram with vertices .
[You may assume that is holomorphic on .]
Now let and be two sets of complex numbers and
Prove that is a doubly-periodic meromorphic function, with periods 1 and , if and only if is an integer.
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B3.9
2004 comment(a) Let be a non-constant holomorphic map between compact connected Riemann surfaces and .
Define the branching order at a point and show that it is well-defined. Show further that if is a holomorphic map on then .
Define the degree of and state the Riemann-Hurwitz formula. Show that if has Euler characteristic 0 then either is the 2 -sphere or for all .
(b) Let and be complex polynomials of degree with no common roots. Explain briefly how the rational function induces a holomorphic map from the 2-sphere to itself. What is the degree of ? Show that there is at least one and at most points such that the number of distinct solutions of the equation is strictly less than .
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B4.8
2004 commentLet be a lattice in , where and . By constructing an appropriate family of charts, show that the torus is a Riemann surface and that the natural projection is a holomorphic map.
[You may assume without proof any known topological properties of .]
Let be another lattice in , with and . By considering paths from 0 to an arbitrary , show that if is a conformal equivalence then
[Any form of the Monodromy Theorem and basic results on the lifts of paths may be used without proof, provided that these are correctly stated. You may assume without proof that every injective holomorphic function is of the form , for some .]
Give an explicit example of a non-constant holomorphic map that is not a conformal equivalence.
Part II 2004
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B2.10
2004 commentFor each of the following curves
(i) (ii)
compute the points at infinity of (i.e. describe ), and find the singular points of the projective curve .
At which points of is the rational map , given by , not defined? Justify your answer.
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B3.10
2004 comment(i) Let be a morphism of smooth projective curves. Define the divisor if is a divisor on , and state the "finiteness theorem".
(ii) Suppose is a morphism of degree 2 , that is smooth projective, and that . Let be distinct ramification points for . Show that, as elements of , we have , but .
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B4.9
2004 commentLet be an irreducible homogeneous polynomial of degree , and write for the curve it defines in . Suppose is smooth. Show that the degree of its canonical class is .
Hence, or otherwise, show that a smooth curve of genus 2 does not embed in .
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B1.13
2004 commentLet be a probability space and let be a sequence of events.
(a) What is meant by saying that is a family of independent events?
(b) Define the events infinitely often and eventually . State and prove the two Borel-Cantelli lemmas for .
(c) Let be the outcomes of a sequence of independent flips of a fair coin,
Let be the length of the run beginning at the flip. For example, if the first fourteen outcomes are 01110010000110 , then , etc.
Show that
and furthermore that
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B2.12
2004 commentLet be a measure space and let .
(a) Define the -norm of a measurable function , and define the space
(b) Prove Minkowski's inequality:
[You may use Hölder's inequality without proof provided it is clearly stated.]
(c) Explain what is meant by saying that is complete. Show that is complete.
(d) Suppose that is a sequence of measurable functions satisfying as .
(i) Show that if , then almost everywhere.
(ii) When , give an example of a measure space and such a sequence such that, for all as .
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B3.12
2004 comment(a) Let be a probability space and let be measurable. What is meant by saying that is measure-preserving? Define an invariant event and an invariant random variable, and explain what is meant by saying that is ergodic.
(b) Let be a probability measure on . Let
let be the smallest -field of with respect to which the coordinate maps , for , are measurable, and let be the unique probability measure on satisfying
for all . Define by for .
(i) Show that is measurable and measure-preserving.
(ii) Define the tail -field of the coordinate maps , and show that the invariant -field of satisfies . Deduce that is ergodic. [Any general result used must be stated clearly but the proof may be omitted.]
(c) State Birkhoff's ergodic theorem and explain how to deduce that, given independent identically-distributed integrable random variables , there exists such that
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B4.11
2004 commentLet be a probability space and let be random variables. Write an essay in which you discuss the statement: if almost everywhere, then . You should include accounts of monotone, dominated, and bounded convergence, and of Fatou's lemma.
[You may assume without proof the following fact. Let be a measure space, and let be non-negative with finite integral If are non-negative measurable functions with for all , then as .]
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B2.13
2004 commentLet be a Poisson random measure of intensity on the plane . Denote by the circle of radius in centred at the origin and let be the largest radius such that contains precisely points of . [Thus is the largest circle about the origin containing no points of is the largest circle about the origin containing a single point of , and so on.] Calculate and .
Now let be a Poisson random measure of intensity on the line . Let be the length of the largest open interval that covers the origin and contains precisely points of . [Thus gives the length of the largest interval containing the origin but no points of gives the length of the largest interval containing the origin and a single point of , and so on.] Calculate and .
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B3.13
2004 commentLet be a renewal process with holding times and be a renewal-reward process over with a sequence of rewards . Under assumptions on and which you should state clearly, prove that the ratios
converge as . You should specify the form of convergence guaranteed by your assumptions. The law of large numbers, in the appropriate form, for sums and can be used without proof.
In a mountain resort, when you rent skiing equipment you are given two options. (1) You buy an insurance waiver that costs where is the daily equipment rent. Under this option, the shop will immediately replace, at no cost to you, any piece of equipment you break during the day, no matter how many breaks you had. (2) If you don't buy the waiver, you'll pay in the case of any break.
To find out which option is better for me, I decided to set up two models of renewalreward process . In the first model, (Option 1), all of the holding times are equal to 6 . In the second model, given that there is no break on day (an event of probability , we have , but given that there is a break on day , we have that is uniformly distributed on , and . (In the second model, I would not continue skiing after a break, whereas in the first I would.)
Calculate in each of these models the limit
representing the long-term average cost of a unit of my skiing time.
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B4.12
2004 commentConsider an queue with . Here is the arrival rate and is the mean service time. Prove that in equilibrium, the customer's waiting time has the moment-generating function given by
where is the moment-generating function of service time .
[You may assume that in equilibrium, the queue size at the time immediately after the customer's departure has the probability generating function
Deduce that when the service times are exponential of rate then
Further, deduce that takes value 0 with probability and that
Sketch the graph of as a function of .
Now consider the queue in the heavy traffic approximation, when the service-time distribution is kept fixed and the arrival rate , so that . Assuming that the second moment , check that the limiting distribution of the re-scaled waiting time is exponential, with rate .
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B1.14
2004 commentState the formula for the capacity of a memoryless channel.
(a) Consider a memoryless channel where there are two input symbols, and , and three output symbols, , *. Suppose each input symbol is left intact with probability , and transformed into a with probability . Write down the channel matrix, and calculate the capacity.
(b) Now suppose the output is further processed by someone who cannot distinguish and , so that the matrix becomes
Calculate the new capacity.
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B2.14
2004 commentFor integer-valued random variables and , define the relative entropy of relative to .
Prove that , with equality if and only if for all .
By considering , a geometric random variable with parameter chosen appropriately, show that if the mean , then
with equality if is geometric.
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B4.13
2004 commentDefine a cyclic code of length .
Show how codewords can be identified with polynomials in such a way that cyclic codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule.
Prove that any cyclic code has a unique generator, i.e. a polynomial of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals , and show that divides .
Let be a cyclic code. Set
(the dual code). Prove that is cyclic and establish how the generators of and are related to each other.
Show that the repetition and parity codes are cyclic, and determine their generators.
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B2.15
2004 commentA gambler is presented with a sequence of random numbers, , one at a time. The distribution of is
where . The gambler must choose exactly one of the numbers, just after it has been presented and before any further numbers are presented, but must wait until all the numbers are presented before his payback can be decided. It costs to play the game. The gambler receives payback as follows: nothing if he chooses the smallest of all the numbers, if he chooses the largest of all the numbers, and otherwise.
Show that there is an optimal strategy of the form "Choose the first number such that either (i) and , or (ii) ", where you should determine the constant as explicitly as you can.
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B3.14
2004 commentThe strength of the economy evolves according to the equation
where and is the effort that the government puts into reform at time . The government wishes to maximize its chance of re-election at a given future time , where this chance is some monotone increasing function of
Use Pontryagin's maximum principle to determine the government's optimal reform policy, and show that the optimal trajectory of is
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B4.14
2004 commentConsider the deterministic dynamical system
where and are constant matrices, , and is the control variable, . What does it mean to say that the system is controllable?
Let . Show that if is the set of possible values for as the control is allowed to vary, then is a vector space.
Show that each of the following three conditions is equivalent to controllability of the system.
(i) The set for all .
(ii) The matrix is (strictly) positive definite.
(iii) The matrix has rank .
Consider the scalar system
where . Show that this system is controllable.
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B1.18
2004 comment(a) State and prove the Mean Value Theorem for harmonic functions.
(b) Let be a harmonic function on an open set . Let . For any and for any such that , show that
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B2.18
2004 comment(a) State and prove the Duhamel principle for the wave equation.
(b) Let be a solution of
where is taken in the variables and etc.
Using an 'energy method', or otherwise, show that, if on the set for some , then vanishes on the region . Hence deduce uniqueness for the Cauchy problem for the above PDE with Schwartz initial data.
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B3.18
2004 comment(i) Find such that is a Schwartz function of for each and solves
where and are given Schwartz functions and denotes . If represents the Fourier transform operator in the variables only and represents its inverse, show that the solution satisfies
and calculate in Schwartz space.
(ii) Using the results of Part (i), or otherwise, show that there exists a solution of the initial value problem
with and given Schwartz functions, such that
as in Schwartz space, where is the solution of
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B4.18
2004 comment(a) State a theorem of local existence, uniqueness and dependence on the initial data for a solution for an ordinary differential equation. Assuming existence, prove that the solution depends continuously on the initial data.
(b) State a theorem of local existence of a solution for a general quasilinear firstorder partial differential equation with data on a smooth non-characteristic hypersurface. Prove this theorem in the linear case assuming the validity of the theorem in part (a); explain in your proof the importance of the non-characteristic condition.
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B1.19
2004 commentState the convolution theorem for Laplace transforms.
The temperature in a semi-infinite rod satisfies the heat equation
and the conditions for for and as . Show that
where
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B2.19
2004 comment(a) The Beta function is defined by
Show that
(b) The function is defined by
where the integrand has a branch cut along the positive real axis. Just above the cut, . For just above the cut, arg . The contour runs from , round the origin in the negative sense, to (i.e. the contour is a reflection of the usual Hankel contour). What restriction must be placed on and for the integral to converge?
By evaluating in two ways, show that
where and are any non-integer complex numbers.
Using the identity
deduce that
and hence that
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B3.19
2004 commentThe function satisfies the third-order differential equation
subject to the conditions as and . Obtain an integral representation for of the form
and determine the function and the contour .
Using the change of variable , or otherwise, compute the leading term in the asymptotic expansion of as .
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B4.19
2004 commentLet . Sketch the path of const. through the point , and the path of const. through the point .
By integrating along these paths, show that as
where the constants and are to be computed.
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B1.21
2004 commentThe Maxwell field tensor is
and the 4-current density is . Write down the 3-vector form of Maxwell's equations and the continuity equation, and obtain the equivalent 4-vector equations.
Consider a Lorentz transformation from a frame to a frame moving with relative (coordinate) velocity in the -direction
where . Obtain the transformation laws for and . Which quantities, quadratic in and , are Lorentz scalars?
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B2.21
2004 commentA particle of rest mass and charge moves along a path , where is the particle's proper time. The equation of motion is
where etc., is the Maxwell field tensor , where and are the -components of the electric and magnetic fields) and is the Minkowski metric tensor. Show that and interpret both the equation of motion and this equation in the classical limit.
The electromagnetic field is given in cartesian coordinates by and , where is constant and uniform. The particle starts from rest at the origin. Show that the orbit is given by
where .
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B4.21
2004 commentUsing Lorentz gauge, , Maxwell's equations for a current distribution can be reduced to . The retarded solution is
where . Explain, heuristically, the rôle of the -function and Heaviside step function in this formula.
The current distribution is produced by a point particle of charge moving on a world line , where is the particle's proper time, so that
where . Show that
where , and further that, setting ,
where should be defined. Verify that
Evaluating quantities at show that
where . Hence verify that and
Verify this formula for a stationary point charge at the origin.
[Hint: If has simple zeros at then
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B1.22
2004 commentDefine the notions of entropy and thermodynamic temperature for a gas of particles in a variable volume . Derive the fundamental relation
The free energy of the gas is defined as . Why is it convenient to regard as a function of and ? By considering , or otherwise, show that
Deduce that the entropy of an ideal gas, whose equation of state is (using energy units), has the form
where is independent of and .
Show that if the gas is in contact with a heat bath at temperature , then the probability of finding the gas in a particular quantum microstate of energy is
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B3.22
2004 commentDescribe briefly why a low density gas can be investigated using classical statistical mechanics.
Explain why, for a gas of structureless atoms, the measure on phase space is
and the probability density in phase space is proportional to
where is the temperature in energy units.
Derive the Maxwell probability distribution for atomic speeds ,
Why is this valid even if the atoms interact?
Find the mean value of the speed of the atoms.
Is the mean kinetic energy of the atoms?
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B4.23
2004 commentDerive the Bose-Einstein expression for the mean number of Bose particles occupying a particular single-particle quantum state of energy , when the chemical potential is and the temperature is in energy units.
Why is the chemical potential for a gas of photons given by ?
Show that, for black-body radiation in a cavity of volume at temperature , the mean number of photons in the angular frequency range is
Hence, show that the total energy of the radiation in the cavity is
where is a constant that need not be evaluated.
Use thermodynamic reasoning to find the entropy and pressure of the radiation and verify that
Why is this last result to be expected for a gas of photons?
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B1.23
2004 commentThe operator corresponding to a rotation through an angle about an axis , where is a unit vector, is
If is unitary show that must be hermitian. Let be a vector operator such that
Work out the commutators . Calculate
for each component of .
If are standard angular momentum states determine for any and also determine .
Hint
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B2.23
2004 commentThe wave function for a single particle with a potential has the asymptotic form for large
How is related to observable quantities? Show how can be expressed in terms of phase shifts for ..
Assume that for , and let denote the solution of the radial Schrödinger equation, regular at , with energy and angular momentum . Let . Show that
Assuming that is a smooth function for , determine the expected behaviour of as . Show that for then , with a constant, and determine in terms of .
[For the two independent solutions of the radial Schrödinger equation are and with
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B3.23
2004 commentFor a periodic potential , where is a lattice vector, show that we may write
where the set of should be defined.
Show how to construct general wave functions satisfying in terms of free plane-wave wave-functions.
Show that the nearly free electron model gives an energy gap when .
Explain why, for a periodic potential, the allowed energies form bands where may be restricted to a single Brillouin zone. Show that if and belong to the Brillouin zone.
How are bands related to whether a material is a conductor or an insulator?
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B4.24
2004 commentDescribe briefly the variational approach to determining approximate energy eigenvalues for a Hamiltonian .
Consider a Hamiltonian and two states such that
Show that, by considering a linear combination , the variational method gives
as approximate energy eigenvalues.
Consider the Hamiltonian for an electron in the presence of two protons at and ,
Let be the ground state hydrogen atom wave function which satisfies
It is given that
and, for large , that
Consider the trial wave function . Show that the variational estimate for the ground state energy for large is
Explain why there is an attractive force between the two protons for large .
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B1.25
2004 commentConsider a uniform stream of inviscid incompressible fluid incident onto a twodimensional body (such as a circular cylinder). Sketch the flow in the region close to the stagnation point, , at the front of the body.
Let the fluid now have a small but non-zero viscosity. Using local co-ordinates along the boundary and normal to it, with the stagnation point as origin and in the fluid, explain why the local outer, inviscid flow is approximately of the form
for some positive constant .
Use scaling arguments to find the thickness of the boundary layer on the body near . Hence show that there is a solution of the boundary layer equations of the form
where is a suitable similarity variable and satisfies
What are the appropriate boundary conditions for and why? Explain briefly how you would obtain a numerical solution to subject to the appropriate boundary conditions.
Explain why it is neither possible nor appropriate to perform a similar analysis near the rear stagnation point of the inviscid flow.
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B2.25
2004 commentAn incompressible fluid with density and viscosity is forced by a pressure difference through the narrow gap between two parallel circular cylinders of radius with axes apart. Explaining any approximations made, show that, provided and , the volume flux (per unit length of cylinder) is
when the cylinders are stationary.
Show also that when the two cylinders rotate with angular velocities and respectively, the change in the volume flux is
For the case , find and sketch the function , where is the centreline velocity at position along the gap in the direction of flow. Comment on the values taken by .
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B3.24
2004 commentUsing the Milne-Thompson circle theorem, or otherwise, write down the complex potential describing inviscid incompressible two-dimensional flow past a circular cylinder of radius centred on the origin, with circulation and uniform velocity in the far field.
Hence, or otherwise, find an expression for the velocity field if the cylinder is replaced by a flat plate of length , centred on the origin and aligned with the -axis. Evaluate the velocity field on the two sides of the plate and confirm that the normal velocity is zero.
Explain the significance of the Kutta condition, and determine the value of the circulation that satisfies the Kutta condition when .
With this value of the circulation, calculate the difference in pressure between the upper and lower sides of the plate at position . Comment briefly on the value of the pressure at the leading edge and the force that this would produce if the plate had a small non-zero thickness.
Determine the force on the plate, explaining carefully the direction in which it acts.
[The Blasius formula , where is a closed contour lying just outside the body, may be used without proof.]
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B4.26
2004 commentWrite an essay on the Kelvin-Helmholtz instability of a vortex sheet. Your essay should include a detailed linearised analysis, a physical interpretation of the instability, and an informal discussion of nonlinear effects and of the effects of viscosity.
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B1.26
2004 commentA physical system permits one-dimensional wave propagation in the -direction according to the equation
where is a real positive constant. Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wave number. Is it the shortest or the longest waves that are at the front of a dispersing wave train arising from a localised initial disturbance? Do the wave crests move faster or slower than a packet of waves?
Find the solution of the above equation for the initial disturbance given by
where is real and .
Use the method of stationary phase to obtain a leading-order approximation to this solution for large when is held fixed.
[Note that
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B2.26
2004 commentThe linearised equation of motion governing small disturbances in a homogeneous elastic medium of density is
where is the displacement, and and are the Lamé constants. Derive solutions for plane longitudinal waves with wavespeed , and plane shear waves with wavespeed .
The half-space is filled with the elastic solid described above, while the slab is filled with an elastic solid with shear modulus , and wavespeeds and . There is a vacuum in . A harmonic plane wave of frequency and unit amplitude propagates from towards the interface . The wavevector is in the -plane, and makes an angle with the -axis. Derive the complex amplitude, , of the reflected wave in . Evaluate for all possible values of , and explain your answer.
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B3.25
2004 commentThe dispersion relation for sound waves of frequency in a stationary, homogeneous gas is , where is the speed of sound and is the wavevector. Derive the dispersion relation for sound waves of frequency in a uniform flow with velocity U.
For a slowly-varying medium with a local dispersion relation , derive the ray-tracing equations
The meaning of the notation should be carefully explained.
Suppose that two-dimensional sound waves with initial wavevector are generated at the origin in a gas occupying the half-space . The gas has a mean velocity , where . Show that
(a) if and then an initially upward propagating wavepacket returns to the level within a finite time, after having reached a maximum height that should be identified;
(b) if and then an initially upward propagating wavepacket continues to propagate upwards for all time.
For the case of a fixed frequency disturbance comment briefly on whether or not there is a quiet zone.
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B4.27
2004 commentA plane shock is moving with speed into a perfect gas. Ahead of the shock the gas is at rest with pressure and density , while behind the shock the velocity, pressure and density of the gas are and respectively. Derive the Rankine-Hugoniot relations across the shock. Show that
where and is the ratio of the specific heats of the gas. Now consider a change of frame such that the shock is stationary and the gas has a component of velocity parallel to the shock. Deduce that the angle of deflection of the flow which is produced by a stationary shock inclined at an angle to an oncoming stream of Mach number is given by
[Note that